MATHEMATICS OF GRAPHING AND SURVEYING CHAPTER 9 INTRODUCTION TO GRAPHING AND SURVEYING

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1 CHAPTER 9 MATHEMATICS OF GRAPHING AND SURVEYING INTRODUCTION TO GRAPHING AND SURVEYING Real estate typically refers to a physical asset that is tied to one geographic location. Because it is durable (lasts a long time) and immobile, determining the exact location and size of a given parcel of land is critical. For example, when the government grants land, when a plot of land is subdivided into individual lots, or when a property is to be developed, the exact location, dimensions, and size of the lot must be specified. The means of determining these specifications is by surveying. This chapter provides an introduction to the kinds of measurements and calculations that a real estate professional must be familiar with in order to understand and use surveys. We will discuss various mathematical concepts and provide practical examples to illustrate their application. The first half of the chapter discusses graphing methods. The chapter then proceeds to discuss the basic fundamentals of surveying and building measurement.

2 9.2 Foundations of Real Estate Mathematics PART ONE: GRAPHING Why Study Graphs? Graphing is a tool that can be used in real estate practice to assist in research and to more effectively present findings. Real estate practitioners often need to gather considerable amounts of data, which must be analyzed and presented in a form that is clear and understandable to all readers. Sometimes the complexity of the data gathered in a research situation makes it difficult to see the interrelationships among the data and therefore makes it difficult to draw conclusions. In these cases, it may be helpful to chart the information on a graph. A graph may be defined simply as a diagram that by means of dots and lines shows a relationship between data. This section introduces how graphs can help analyze and present mathematical data. Presentation is an important part of any problem analysis, because something that may be simple or self-evident to the writer of a report may not seem so simple to the reader. Graphs and Formulas All information that can be represented by a mathematical formula may be represented in a graph form. For example, any number can be represented as a point on a number line in order to illustrate the number visually. Figure 9.1 illustrates a timeline used for investment analysis, where each point on the line graph may represent one year, thus displaying an investment pattern over a series of time. FIGURE 9.1 Timeline Example PV = $10,000 PMT = 0 j 1 = 5% years FV =? If you wish to visually represent a pair of numbers that belong together, you can do so by means of a two-dimensional graph. For example, suppose that you have found a parcel of land with 40 frontage feet that recently sold for $86,500. Both of these numbers have a distinct meaning, but they offer much more useful information when they are provided together. They can be visually represented by extending the idea of a number line to two dimensions. A two-dimensional graph is created by drawing two number lines at right angles to each other (one horizontal and one vertical), with the lines crossing at their respective zero points. A given pair of numbers can then be represented by a point in the plane defined by these two lines. The horizontal position corresponds to the first number and the vertical position corresponds to the second, and thus the pair of numbers has a defined location on the graph (see Figure 9.2). FIGURE 9.2 A Two- Dimensional Graph Y-axis Pair of Numbers (x,y) Origin X-axis

3 Chapter 9: Mathematics of Graphing and Surveying 9.3 This graphical representation of pairs of numbers is called the co-ordinate plane or the Cartesian plane (after the French mathematician René Descartes). The perpendicular number lines are called the axes, with the horizontal one usually called the X-axis and the vertical one the Y-axis. The point where the two axes cross, which is designated zero on both lines, is called the origin. An appropriate scale is selected along each axis, with positive numbers increasing to the right along the X-axis and upward on the Y-axis. For example, for the parcel of land described previously, the Y-axis might be used to represent cost in number of dollars, while the X-axis might represent frontage in number of feet. Note that the scale does not have to be the same for both axes. These two axes allow you to visually represent the location of a pair of numbers. When a pair of numbers is graphed, the first number is called the x co-ordinate, because it measures its position along the X-axis, and the second number is called the y co-ordinate because it measures the point s position along the Y-axis. The location of the point corresponding to a pair of numbers (x,y) is then found by counting from the origin x units to the right along the x-axis and then y units up parallel to the Y-axis. For example, Figure 9.3 shows the point P that would correspond to a pair of numbers (6,5) Y-axis P (6,5) FIGURE 9.3 Point Corresponding to a Pair of Numbers X-axis Note that the number pairs (6,5), (5,6), (-6,5), (6,-5), and (-6,-5) are all different and would result in different locations on the graph. If either of the numbers in a pair is negative, then we count to the left or down from the origin instead of to the right or up. For the land sale example discussed previously, the observation of a 40-foot lot costing $86,500 could be represented by locating (40, 86500) on a graph with number of dollars on the Y-axis and number of frontage feet on the X-axis. The origin is set at zero for both axes, with the scale on the Y-axis set as one square = $10,000 and the scale on the X-axis set as one square = 10 feet. This property could then be graphed as shown in Figure (40,86500) FIGURE 9.4 Plotting Land Value by Frontage Foot 70 Dollars (Thousands) Frontage (feet)

4 9.4 Foundations of Real Estate Mathematics A Simple Algebraic Formula A formula using two variables, such as y = x + 2, assigns a value to a variable y given the value of a variable x. For example, if x = 0, this equation says that y = 2; if x = 4, then y = 6; if x = 6, y = 8; if x = -2, y = 0; and so on. By choosing values for one variable and applying the formula to calculate the corresponding value of the other variable, you are using the equation to generate a set of pairs of values (x,y) that satisfy the equation. Since it has been determined that pairs of numbers can be graphed on a plane, you could represent the equation graphically by graphing all the points corresponding to the pairs of values (x,y) that the equation generates. Of course, since you can choose any value for x, the number of pairs defined by an equation is infinite. The best place to start is to list a few pairs, graph them, and look for a pattern in the graphed points. Looking at the equation y = x + 2, we have already found several pairs of values. These are listed below and then graphed in Figure 9.5. X Y FIGURE 9.5 Graphing Pairs of Values 10 8 Y-axis X-axis For another example, consider the following table of census data for Canada from 1901 to 2014 which is then graphed in Figure 9.6, showing population over time. Year Population ,494, ,919, ,654, ,238, ,820, ,021, ,739, ,427,524

5 Chapter 9: Mathematics of Graphing and Surveying ,000,000 35,000,000 30,000,000 FIGURE 9.6 Population of Canada Population 25,000,000 20,000,000 15,000,000 10,000,000 5,000, Year When the results are examined in table form, it looks like population increases at a steady rate. However, the graph indicates that the rate of population growth is actually increasing over time the line is steepening as it moves to the right. Slope-Intercept: Y=mX +b Any equation can be graphed by making a table of values showing several points satisfying the equation and joining the points in a smooth line or curve. Except for the last example, the equations you have worked with so far are called linear equations. The graph of every linear equation is a straight line. Equations are non-linear when they have exponents other than 1, 2 which means that these equations will graph as curves, e.g., try graphing the equations y=x or y= x. There are two properties of straight lines that make them particularly easy to graph: 1. Any two points define a line. In other words, once you recognize that an equation to be graphed is linear, you know that only two points need be plotted to be able to draw the line. Of course, it is wise to check your calculation of the first two points by plotting a third point and confirming that it lies on the same line, but two points are enough. 2. The slope of a straight line. The slope of a line is defined as the change in the y co-ordinate per unit change in the x co-ordinate as you move to the right along the line. Another way to express this definition is rise over run. The slope is a numerical measure of how steep a line is. A straight line has the same steepness everywhere. So we say that its slope is constant. The slope is illustrated in the graph in Figure 9.7 where the line for the equation y = x + 2 is illustrated. The slope of this line is 1, which says that if x increases by 1 unit, y will increase by 1 unit as well. However, if the slope of this line was 3, it would mean that for every 1 unit increase in x, there will be a 3 unit increase in y. Another point of interest is the value of y where x = 0. This value is known as the Y-intercept, because it is the point where the line crosses the Y-axis. For this equation, the Y-intercept is 2. A standard form of expressing a linear equation is y = mx + b, where x represents the numbers given and y represents the results formed from applying the equation. In this standard equation, m is the slope of the line and b is the Y-intercept. This form is called the slope-intercept form of the linear equation, because it permits us to immediately read the values of the slope and y-intercept. For example, in the formula y = x + 2, the slope was 1 and the Y-intercept was 2. Another example of a slope-intercept equation could express selling price (P) in terms of frontage feet (F), P = 2100 F This equation tells us that the total cost of a lot increases by $2,100 per additional frontage foot, while the Y-intercept tells us that if it were possible to buy a zero-frontage lot, the cost would be $2,500, i.e., the lawyer s fee. The graph of this equation has a slope of 2100 and a Y-intercept of 2500 (try graphing it on your own to confirm this). EQUATION 9.1

6 9.6 Foundations of Real Estate Mathematics FIGURE 9.7 Graph of y = x Y-axis X-axis The following illustration shows how raw data might be analyzed and presented in graph form. ILLUSTRATION 9.1 Chris is appraising a small apartment building. He has found the following three recent sales of properties very similar to the subject property: Sale # Subject Sale Price $600,000 $500,000 $250,000? Determine the value of the subject property using the median, 1 the mean, and graphing. Solution: To find the value of the subject property, the most simple solution would be to use the median (middle value) of $500,000 or to use the average (mean) price of $450,000 [($600,000 + $500,000 + $250,000) ) 3]. However, further analysis indicates that this course of action would be inappropriate since one variable alone for the comparable properties (sale price) does not provide enough information to calculate the value of the subject property. Chris has determined that the size of the apartment building is probably an important factor in determining its value, so he has counted the number of suites in each building, with the following results: Sale # Sale Price # of Units Subject $600,000 $500,000 $250,000? There appears to be a relationship here, but it is still not clear. The following graph of sale price by number of units may clarify this relationship. 1 Chapter 10 discusses the median and mean in detail.

7 Chapter 9: Mathematics of Graphing and Surveying 9.7 Sale Price ($1,000) 700 FIGURE 9.8 Sale Price by Number of Units Number of Units The situation becomes clearer after graphing there is definitely a relationship between sale price and number of units, although it may not be a linear one (because the line seems to curve). Further analysis appears to be necessary. Chris has now calculated the sale price per unit: Sale # Sale Price # of Units Sale Price per Unit 1 $600, $60,000 2 $500,000 7 $71,429 3 $250,000 3 $83,333 Subject? 6? Another graph is drawn below to illustrate the relationship between sale price per unit and number of units: FIGURE 9.9 Sale Price per Unit ($1,000) Sale Price per Unit by Number 90 of Units Number of Units

8 9.8 Foundations of Real Estate Mathematics This graph makes it clear that there is a linear relationship between sale price per unit and the number of units. It seems that as apartment buildings get larger, the sale price per unit decreases. Is this reasonable? Some possible opinions: 1. Yes the larger the property, the higher the overall price, and the lower the number of people who can afford to purchase. (demand side) 2. Yes larger buildings have economies of scale with respect to construction costs and operating costs, e.g., a 2 storey or 10 storey building both require the same expense for roofing and for lobby maintenance (supply side). 3. No can you think of any reasons why this would not normally be the case? From this graph, the appraiser can provide convincing evidence that a price per unit of $74,000 would not be unreasonable for valuing the subject property. Given that the subject property has six suites, this would give an overall market value of $444,000. Graphing Summary So far in this chapter, we have considered two approaches to graphing: (1) plotting the observed values of two variables as points and (2) graphing exact relationships between two variables specified by equations as lines or curves. When you are evaluating real data, you will almost always generate the former type of graph. In real estate practice, you will often need to plot graphs of values that you have observed, and will rarely find that they lie precisely on a straight line or a smooth curve. Observing the data points and looking for trends in the data that can be approximately represented by equations or smooth curves involves regression analysis, which will be discussed later in this course. There is a natural relationship between graphing and mapping. The representation of pairs of numbers on a two-dimensional graph is akin to measuring and mapping the boundaries of a property along the cardinal directions of the compass. In the next section of this chapter, we will see that skill with graphing and the relation between points and numbers will allow you to draw scale sketches of properties much more easily and accurately. EXERCISE 9.1 (a) Determine the co-ordinates of each point on the graph below. 8 F 4 E G C A H D B K -4 J I -8

9 Chapter 9: Mathematics of Graphing and Surveying 9.9 A = G = B = H = C = I = D = J = E = K = F = (b) Consider the following pairs of observations: (i) (12,0) (v) ( 4,4) (ii) ( 6,0) (vi) ( 4, 4) (iii) (4,4) (vii) (0, 3) (iv) (4, 4) (viii) (0,3) Graph each of these points on the graph in (a). Solution: (a) A = 2, 2 G = 9, 4 B = 2, 2 H = 13, 0 C = 2, 2 I = 0, 7 D = 2, 2 J = 7, 6 E = 6, 4 K = 12, 3 F = 7, 3 8 (v) (iii) 4 (viii) (ii) (i) (vii) (vi) (iv) -4

10 9.10 Foundations of Real Estate Mathematics EXERCISE 9.2 In performing an appraisal of a lakeside cottage lot, the following comparables were found. Plot these on a graph of frontage versus selling price. X Frontage (m) Y Selling Price 10 $190, $260, $400, $440, $520, $590,000 Solution: Sale Price (Thousands) Frontage (Metre) EXERCISE Graph the following linear equations: (a) y = x + 7 (d) y = x (b) 2y = 3x 6 (e) 2x + y = 10 (c) y = x (f) x + y = 10

11 Chapter 9: Mathematics of Graphing and Surveying 9.11 Solution: (a) y = x (b) 2y = 3x (c) y = x (d) y = -x (e) 2x + y = 10 (f) x + y = PART TWO: SURVEYOR S MATHEMATICS Why Study Surveyor s Mathematics? Instead of the imposing-sounding title Surveyor s Mathematics, perhaps a simpler title that may suffice could be the mathematics of area. In real estate practice, determining area is important, whether you are faced with determining the size of the subject building, the size of a subject lot, or the comparability of a recent sale or listing. Real estate practitioners need to know pertinent facts about the size and shape of properties. For the most part, this information is readily available from surveyors, but some further analysis may be necessary. In introducing this topic, keep in mind there is NOT any intention to teach students how to be expert surveyors; instead, the intention is to teach students to be able to glean

12 9.12 Foundations of Real Estate Mathematics certain information from plans and sketches, and to know when an expert more experienced and qualified might be required. This section of the chapter will first address measurement issues, such as types of measurement (olde English, Imperial, and metric), and then show how to calculate the area of various geometric shapes. The chapter concludes with a description of the basics of mapping and surveying. Basic Measurement Concepts Before looking at survey systems and actual plans (called plats in some areas), we need to explain some basic units of measurement and geometry techniques. Units of Measurement There are many different ways to measure the area of land and buildings, ranging from metric, to Imperial, to olde English. Surveys from the Land Title Office or most municipal offices will show measurements in metric. The information from most Real Estate Boards, such as house or lot size, will be in Imperial measures, such as feet, inches, and acres. Some very old surveys may use rods and chains, an archaic system known as olde English. Imperial and metric measures will likely be familiar to most people, since these are often used interchangeably in everyday life in Canada. The olde English system may not be so familiar. When Canada was originally surveyed, the basis of measurement was the 66 foot long surveyor s chain, which consists of 100 links or 4 rods (also called rood, pole, or perch ). Historically, this explains why standard road allowances are 66 feet wide in many parts of Canada and why road allowances are often separated by a mile (80 chains) or a mile and a quarter (100 chains). Table 9.1 shows different measurements and how to convert between them. This table may be helpful for future reference when reviewing surveys in real estate practice. TABLE 9.1: Measurements and Conversions Olde English 1 link = 7.92 inches 1 rod = 1 perch = 1 pole = 16.5 feet 1 chain = 66 feet = 4 rods = 100 links 1 mile = 5,280 feet = 1,760 yards = 320 rods = 80 chains 1 acre = 43,560 square feet = 10 square chains = hectares 1 square mile = 640 acres Imperial 1 foot = metres 1 inch = 2.54 centimetres 1 square foot = square metres Metric 1 metre = 100 centimetres = feet 1 centimetre = inches = feet 1 square metre = square feet 1 hectare = acres

13 Chapter 9: Mathematics of Graphing and Surveying 9.13 Area of a Rectangle A rectangle is defined as an area bounded by four straight lines, of which two sets are parallel (run in the same direction) and there is at least one right angle. For example: 100' 30' This figure is 30 feet by 100 feet. The 2 lines indicate that both the right and left sides of the figure (e.g., east and west) are equal length of 100 feet. Therefore, the north boundary must be 30 feet (you can verify this by measuring it). The m indicates that the north-west corner is a right angle. The equal lines and the right angle together show this figure must be a rectangle. The formula for calculating the area of a rectangle is: A = L W Or, in other words, length multiplied by width equals area. Referring back to the example above, the area would be: 30' H 100' = 3,000 square feet This formula is useful for finding the area of a square parcel of land or for calculating the living area of a house. Area of a Right-Angled Triangle A triangle is defined as a geometric figure with three straight sides. By definition, the three interior angles must add up to 180E (try measuring the angles on any triangle to verify this). A right-angled triangle has one interior 90E angle. It follows from the 180E rule discussed above that the sum of the other two angles must equal 90E. Pythagoras Theorem In approximately 500 B.C., the Greek mathematician Pythagoras proved a useful relationship for triangles. He found that in right-angled triangles, the sum of the squares of the two shorter sides equals the square of the hypotenuse (longest side). Or, in other words, c 2 = a 2 + b 2. EQUATION 9.2 a c b c = a + b EQUATION 9.3

14 9.14 Foundations of Real Estate Mathematics Pythagoras theorem is useful for finding the length of a line in a right-angled triangle, given the length of the other two lines. The following example illustrates this: What is the length of the hypotenuse for the following triangle? 3? 2 2? = 3 + 4? = 9+ 16? = 25? = 5 4 In this example, Pythagoras Theorem is applied to find that the length of the hypotenuse is 5. This would be useful to a land surveyor who needed to find the length of the third side of a pie-shaped lot, without needing to measure that side. The formula for calculating the area of a triangle is one-half of the base (b) of the triangle times its height (h): EQUATION 9.4 A = ½ b h For example, assume that the triangle given above is a triangular-shaped parcel of land that is 3 miles by 4 miles (with a 5 mile hypotenuse, although this measurement is not needed for this calculation). The area would be: Area = 2 H 3 miles H 4 miles = 6 square miles Other Triangles and Irregular Shapes Triangle: A three-sided polygon h h b b Area = 2 H b H h The altitude (h) of a triangle is the perpendicular distance from any vertex to the opposite side or its extension. Parallelogram A quadrilateral having its opposite sides parallel. Area = b H h The altitude (h) of a parallelogram is the perpendicular distance between the parallel sides.

15 Chapter 9: Mathematics of Graphing and Surveying 9.15 h b Trapezoid A quadrilateral having two and only two sides parallel. Area = h H 2 (b + c) c h Irregular Polygon b The area of irregular polygons can be determined by dividing the area into the shapes previously mentioned and adding the area of the parts. Trapezoid b d a c Triangles e Area = 2 (a H b) + c H 2 (b + d) + ½ (e d)

16 9.16 Foundations of Real Estate Mathematics EQUATION 9.5 Area of a Circle The basic formula for calculating the area of a circle is pi (π) times the square of the radius: A = πr 2 The radius is one-half of the diameter of a circle the diameter is a straight line through the circle s centre from one side to the other. Pi, represented by the Greek letter π, is a commonly used number when measuring the size of circles. Pi is approximately , which is the ratio of the circumference of a circle (measurement around the outside perimeter) to its diameter. The following example illustrates how this formula can be used to calculate the area of a circle. EXAMPLE 9.1 Consider the following diagram of a sun room adjoining the side of a house. What is the area of the semi-circular sun room? 2m 8m 20m 2m Solution: A = πr 2 A = = 201 square metres The area of this entire circle would be 201 square metres (m 2 ). However, we are looking for the area of only one-half of this circle. Therefore, the area of the sun room is 201 m 2 ) 2 = m 2. The examples in Figure 9.10 (and Appendix 9.1) illustrate common residential building types by number of floors, and demonstrate square metre calculations of living areas. In practice, it may not be necessary to spend time measuring a building and estimating the area of its complicated shape if there are registered plans or surveyor s records available. Many buildings, particularly those which have been stratified (i.e., changed to condominium ownership), will have plans registered showing the dimensions and areas of the site and each individual building. You may be able to obtain a copy of these plans from the local Land Title Office or Land Registry for a modest fee.

17 Chapter 9: Mathematics of Graphing and Surveying 9.17 Example 1: One Storey Basically self-explanatory. Exterior length times width of base of dwelling, and exterior length times width of projections and cantilevered areas. Finished basement has same dimensions as Area A. ABOVE GRADE FINISHED FLOOR AREA: Area A: x 7.65 = 86.5m 2 Area B: 4.25 x 1.50 = 6.4m 2 Total = 92.9m 2 + (JOG) & BASEMENT 7.65 FIGURE 9.10 Examples of Living Area and Calculations BELOW GRADE FINISHED FLOOR AREA: Basement: x 7.65 = 86.5m 2 Total = 86.5m 2 BASEMENT Example 2: One and One-Half Storey without Dormers Area A is calculated as exterior length times width of base of dwelling, and exterior length times width of projections and cantilevered areas. Finished basement has same dimensions as Area A. For Area B measurement, consider floor area as interior length times width to knee wall (angle where wall meets ceiling; ceiling height must be greater than 5 feet; at least 50% of space must have height greater than 7 feet). An additional 15 to 20 centimeters may be included to account for the exterior wall. ABOVE GRADE FINISHED FLOOR AREA: Area A: x 7.65 = 78.8m 2 Area B: x 3.95 = 40.7m 2 Total = 119.5m 2 BELOW GRADE FINISHED FLOOR AREA: Basement: x 7.65 = 78.8m 2 Total = 78.8m 2 BASEMENT MAIN FLR. & BASEMENT nd FLR Example 3: One and One-Half Storey with Dormers Area A is calculated as exterior length times width of base of dwelling, and exterior length times width of projections and cantilevered areas. For Area B measurement, consider floor area as interior length times width to knee wall (angle where wall meets ceiling; ceiling height must be greater than 5 feet; at least 50% of space must have height greater than 7 feet), plus length times width of dormers. An additional 15 to 20 centimeters may be included to account for the exterior wall. Finished basement has same dimensions as Area A. ABOVE GRADE FINISHED FLOOR AREA: Area A: x 7.65 = 78.8m 2 Area B: x 3.95 = 40.7m 2 Area C & D: 2(2.12 x 1.48) = 6.3m 2 Total = 125.8m MAIN FLR. & BASEMENT nd FLR C 2.12 BELOW GRADE FINISHED FLOOR AREA: Basement: x 7.65 = 78.8m 2 Total = 78.8m 2 BASEMENT 3.95 D 2.12

18 9.18 Foundations of Real Estate Mathematics FIGURE 9.10 Examples of Living Area and Calculations Example 5: Two Storey with Built-In Garage Area A is the exterior length times width, excluding the garage. Area B is the exterior length times width. Finished basement has same dimensions as Area A. ABOVE GRADE FINISHED FLOOR AREA: Area A: 7.65 x = 82.2m 2 Area B: 7.65 x = 116.7m 2 Total = 198.9m nd FLR BELOW GRADE FINISHED FLOOR AREA: Basement: 7.65 x = 82.2m 2 Total = 82.2m 2 PLUS BUILT-IN GARAGE Built-in Garage: 7.65 x 4.50 = 34.4m 2 Total = 34.4m 2 BASEMENT & BASEMENT MAIN FLR. & BELOW GRADE FINISHED FLR BUILT-IN GARAGE 7.65 Example 6: Two Storey with Bay Window; 2nd Floor Overhang and Side Addition on Main Floor Area A is the exterior length times width. Area D is the exterior length times width, including overhang. Exterior dimensions of addition and bay window (bay window extends to grade). Finished basement has same dimensions as Area A. ABOVE GRADE FINISHED FLOOR AREA: Area A: x = 109.2m 2 Area B: 0.55 x 2.05 = 1.1m 2 Area C: 4.25 x 1.85 = 7.9m 2 Area D: x = 117.1m 2 Total = 235.3m 2 BASEMENT BELOW GRADE FINISHED FLOOR AREA: Basement: x = 109.2m 2 Total = 109.2m AREA C (MAIN FLR. ONLY) & BASEMENT MAIN FLR. & BELOW GRADE FINISHED FLR nd FLR. OVERHANG 2.05 AREA D 2nd FLR. INCLUDES OVERHANG 0.75 Figure 9.11 illustrates a sample surveying plan. The plan shows (1) Lot A is a corner lot; (2) its length and width; (3) its size; and (4) other characteristics such as the larger lot from which Lot A came. Basic Trigonometry In the plan shown in Figure 9.11, the surveyor calculated the area of Lot A for us, hectares. What if the surveyor had not given the size? Say a rough estimate is required immediately (not an uncommon occurrence, with most clients consistently demanding immediate results). If the property is a rectangle or circle, this is easy to calculate, given the formulas shown earlier. However, properties are typically irregularly shaped. To find the size of these, you must be able to estimate the area under a curve and use basic trigonometry. Note that most real estate professionals will get through their career without needing to resort to this type of mathematics. The vast majority of land in Canada is described using lot and block or registered plans, which means that they have already been surveyed and the sizes and shapes recorded. Given the limited application of trigonometry to most real estate practitioners, this material will not be tested on the assignment or final exam for this course. Individuals who service rural areas where metes and bounds and similar types of legal descriptions are still found may find trigonometry useful.

19 Chapter 9: Mathematics of Graphing and Surveying 9.19 FIGURE 9.11 Sample Surveying Plan

20 9.20 Foundations of Real Estate Mathematics EXERCISE 9.4 Calculate the size of the following areas: CALCULATIONS Hint: you will need to use deductive reasoning to find some required data, such as the measurements for some lines Carport Deck

21 Chapter 9: Mathematics of Graphing and Surveying 9.21 Solution: CALCULATIONS 10 (A) 25 (B) 20 A = 25 H 30 = 750 B = 10 H 25 H 0.5 = 125 C = 20 H 27 = 540 Total = 1, (C) 27 Hint: To find the 10 at the top of the triangle in Area B, you have to find 40! 30 = 10. This form of reasoning is necessary for several measurements in the figures below (A) 8 40 (E) (B) (C) 20 (D) 9 15 Carport A = 8 H 12 = 96 B = ( ) H (20 +9! 15) = 658 C = 15 H 15 = 225 D = 5 H 9 = 45 E = (12 H 11) + (8 H 11 H 0.5) = 176 Total = 1,200 Deck: 8 H 11 H 0.5 = 44 Carport: 20 H 20 = Deck 8 (E) (D) 5 30 (F) (C) (B) 20 (A) A = 15 H 15 = B = (35! 15) H (15 H 12.5) = C = 30 H (55! 5) = 1, D = 5 H (30! 15) = E = (7.5 H 7.5 H ( ) H 0.5 = F = (15 H 5) + (5 H 5) + (0.5 H 5 H 5) + (0.5 H 5 H 5) = Total = 2, Rounded to 2,538 15

22 9.22 Foundations of Real Estate Mathematics FIGURE 9.12 Quadrants of a Surveyor s Compass Basic Survey Concepts The next section describes basic survey systems found across Canada: metes and bounds, section and township systems, and lot and block survey (registered plans). Appraisers regularly make reference to surveys, measurements, and areas, and rely on these facts in appraising the value of properties. Therefore, there is a need for an appraiser to understand the elementary fundamentals of surveying and be able to interpret legal surveys. However, appraisers must also be careful to qualify any estimate of land size by suggesting a survey be carried out by someone qualified to do so. An appraiser is not a surveyor and should not attempt to act as one. Please keep this in mind when dealing with survey plans that are old or complex (or both). Also keep in mind that advanced surveying math, particularly that including trigonometry, likely goes beyond most real estate practitioner s expertise. Using Legal Descriptions to Determine Site Location The legal description of land is based on surveys. These may be purely visual and highly subjective or precise mathematical delineations of the limits of the land. The land description systems commonly in use are metes and bounds, section and township, and lot and block (registered plans). Most registered plans indicate the size and dimensions of the site, and portions of a property surveyed by section/township can generally be determined with relative ease. In terms of needing to calculate land areas, the real estate practitioner s main concern is with metes and bounds description of irregular sites. Metes and Bounds The earliest form of land description was the bounded description, which described the property by reference to physical features or adjacent property owners. Typical physical features included trees, ridge lines, streams, and roads. For example, property might be described as bounded on the north by French Creek, bounded on the east by the land of Ezra Jones, bounded on the south by a wooden fence, and bounded on the west by a line of trees and the county road. Little thought was given to the fact that a fence line or tree might someday cease to exist. As land became more valuable and disputes over unclear bounds became more numerous, better methods of description evolved. It became a common surveying practice to measure the direction of property lines with compass bearings and the distance with measuring chains or tapes. Property described in this manner became known as metes and bounds. The term metes refers to measurements and the term bounds refers to the boundaries, including features of the terrain described in conjunction with compass bearing and distances. The bearing of a line is the angle between a north-south meridian and the line measured from north or south toward the east or west. As illustrated in Figure 9.12, property lines are described in terms of direction within one of the four quadrants of the compass - northeast, southeast, southwest, northwest (NE, SE, SW, NW). 0 N 45 Line bearing N 45 E (Northeast quadrant) 90 Northwest quadrant (NW) Northeast quadrant (NE) 90 Southwest quadrant (SW) Southeast quadrant (SE) S 0

23 Chapter 9: Mathematics of Graphing and Surveying 9.23 N Main north row FIGURE 9.13 Plot of Metes and Bounds Description south row 300' 284' point of beginning 210' Cook Maple east row 251.4' Smith 134.5' 197.3' Wilson In a metes and bounds description, each property line is described with the beginning of each line being the end of the preceding line. A proper metes and bounds description should close ; that is, the last line should come back to the point of beginning. A metes and bounds survey is described below and illustrated in Figure Beginning at an iron pin located in the South Right of Way of Main Street, Said iron pin is located N 88 degrees 15 minutes E at a distance or 284 feet from the intersection of the South ROW of Main Street and the East ROW of Maple Street. Said point is also located at the Northwest corner of a lot now owned by John Smith, Thence leaving said ROW and running with Smith s line S 5 degrees 34 minutes E a distance of feet to an iron pin at the corner of Wilson; thence along Wilson s North line S 82 degrees 41 minutes E a distance of feet to a point in Cook s West line, thence N 46 degrees 10 minutes E a distance of feet, thence N 5 degrees W a distance of 210 feet to an iron pin in the South ROW of Main Street, thence along the South ROW of Main Street S 84 degrees 50 minutes W a distance of 300 feet to the point of beginning, consisting of 1.65 acres, more or less. Section and Township System The section and township or rectangular survey system covers British Columbia, Alberta, Saskatchewan, Manitoba, and northwestern Ontario. These systems have a number of independent points of origin through which pass true meridians of longitude and parallels of latitude, called, respectively, principal meridians and base lines. A principal meridian is a true northsouth line that passes through the geographic poles of the earth. A base line is a line that runs east and west parallel to the equator. In Canada, the principal meridian is located approximately 12 miles west of Winnipeg and the base line is the 49 th parallel. Units of land approximately six miles square are established north of the base line and east and west of the principal meridian. Each six-mile division north of the base line is called a township, and each six-mile division east and west is called a range. Each six-mile-square unit is also called a township (Figure 9.14). A township contains approximately 36 square miles and is divided into 36 sections, each approximately one mile square (or approximately 640 acres). North 3 meridian West Base line 1 East T2S 2 Initial point R3W 3 Principal 2 FIGURE 9.14 Rectangular Survey System: Township Illustration South

24 9.24 Foundations of Real Estate Mathematics In Canada, the sections are numbered consecutively, beginning in the southeast corner of the township and continuing west to the southwest corner, then east, then west and so on until each of the 35 sections is numbered, with section 36 in the northeast corner (Figure 9.15). In the United States, the numbering system begins as the northeast corner of the township and ends in the southeast corner. Road allowances of 66 feet or 1 chain are allowed along every alternate section line running east and west. Thus, in most subdivisions there is a road along the south and north boundaries of a township and along the second and fourth sections lines north of the south boundary of the township. FIGURE 9.15 Section Numbers An ideal section would be one mile square (640 acres). It may be subdivided into quarter sections (160 acres) or quarter-quarter sections (40 acres). However, because the meridians converge toward the poles, it is impossible for townships to be perfectly regular. In addition, surveyors often made mistakes in laying out township and section lines. The discrepancies from either cause are concentrated along the west and north sides of each township. Quarter-quarter sections that do not have the standard 40 acres are called fractional lots or government lots, whether the irregularity is caused by survey or error, rivers and lakes, First Nations land boundaries, provincial boundaries, or any other reason (Figure 9.16). FIGURE 9.16 Fractional Lots NW ¼ 2 NW ¼ Navigable stream SE ¼ 9 SW ¼ (meanderable) 6 1 E ½ NE ¼ Nonnavigable stream SE 1/4 NW ¼ NW ¼ SW ¼ NW ¼ NE ¼ NW ¼ SE ¼ NW ¼ N ½ SW ¼ Lot 1 Lot 2 Lake Northeast quarter (NE ¼) West half of southeast quarter E ½ SE ¼ The regular subdivision of sections is done by reference to halves and quarters. Although sections may be subdivided into units of 2.5 acres (1/256 th of a section), 10-acre units quarter-quarter-quarter sections are usually the smallest. In describing land, the smallest unit is given first and largest last (for example, the S 2 of the NE 3 of the NW 3 of Section 14, Township 2 North, Range 3 West ). In locating property, the description is read backwards, from the largest to the smallest unit. For example, a legal description could be given as Section 15, Township 35, Range 10, west of the 4 th meridian. This is shortened to Sec. 15 Twp 35-Rge 10 W4 th, or Sec. 15/35/10W4. For a quarter section within this section, the description could be SW 3 Sec. 15/35/10-W4. There is another refinement of the subdivision of a section of land. This is a method where the section is divided into 16 parcels of 40 acres, each being called a Legal Subdivision or LSD.

25 Chapter 9: Mathematics of Graphing and Surveying 9.25 The numbering is consecutive starting in the southeast corner and going west to number 4, then one row north, then east, and continuing east to west, west to east, to number 16 in the northeast corner. This method is sometimes used by farmers, and is common in the oil industry and in municipal work. It eliminates the confusion of such descriptions as the NE 3 of the SW 3 of section 15. This would be LSD 6 Sec. 15. Lot and Block Survey Small units of land, particularly in urban areas, are commonly described by lot and block survey. This is a map in which a larger parcel of land is subdivided into small units for the purposes of sale. This map is recorded in the office of the recorder of deeds or land titles. Although each lot is actually surveyed with a metes and bounds description, conveyances need refer only to the lot, block, and map book designation. It is not necessary to include the survey bearing and distances. For example, the land description could simply state: Lot 7, Block 2, Map Book 35, Page 52, or Lot 10, Cameron s Subdivision (and then typically the recorder s office and district in which recorded). Figure 9.17 shows a page from a map book. N00 27'50" W N00 27'50" W Vegas Avenue N89 55'50" W Watson Street 6 50 N00 27'50" W N00 27'50" W N15 05' E R = R = 20 N14 32'12" E Idena Avenue Virgil Street FIGURE 9.17 Example of Lot and Block Description Tract 1523 (Map Book 35, page 52) Rectangular Coordinates Most maps are drawn on flat paper, but the earth has a curved surface. When small areas are mapped, the earth is projected onto a flat plane. For large areas, however, it is necessary to take the earth s curvature into account. A coordinate system creates a plane grid system of rectangular coordinates expressed in feet or metres. The purpose of this coordinate system is to describe points by the use of perpendicular x and y axes. As shown in Figure 9.18, a point of origin is located south and west of the area to be covered by the system. Any point can be described by reference to its distance east of the point of origin along the x and north of the point or origin along the y axis. The actual location of the point is the intersection of lines drawn perpendicular to the x and y axes. North y-direction in feet y-axis 1,500 1, x-axis ,000 1,500 East x-direction in feet Description by coordinates: Point 1: x = 1,200, y = 500 Point 2: x = 300, y = 1,100 Point 3: x = 1,000, y = 1,500 Point 4: x = 1,500, y = 1,300 Point 1: x = 1,200, y = 500 Coordinates converted to bearings and distances: Point 1 to Point 2: bearing N W, distance 1, feet Point 2 to Point 3: bearing N E, distance feet Point 3 to Point 4: bearing S E, distance feet Point 4 to Point 1: bearing S W, distance feet FIGURE 9.18 Description of Land using State Plane Coordinate System

26 9.26 Foundations of Real Estate Mathematics FIGURE 9.19 Location of Prime Meridian, Latitude, and Longitude A plane coordinate system may use the terms eastings and northings instead of x and y. A parcel of property can be described by giving the coordinates of each corner. It is also possible to convert coordinates to a description with bearings and distances. Geographic Coordinates One of the oldest systematic methods of locating a point on the earth is the system of latitudes and longitudes. The distance of a point north or south of the equator is its latitude. The distance of a point east or west of the prime meridian is its longitude. Figure 9.19 shows a globe that has been divided into north-south meridians and parallels of latitude. The prime meridian, from which most countries measure longitude, is the meridian running through Greenwich, England, a suburb of London. N W Longitude E Latitude Equator Prime Meridian S Latitude and longitude are measured in degrees, minutes, seconds, and, if necessary, decimals of a second. At any point on the earth, the ground distance covered by a degree of latitude is approximately 111 kilometres or 69 miles. One second of latitude is therefore approximately 30 metres or 100 feet. The ground distance covered by one degree of longitude is also 111 km at the equator, but it decreases towards the north and south, becoming zero at the poles. For example, at Winnipeg (latitude 49 degrees, 53 minutes) one second of longitude equals m. At Inuvik (latitude 68 degrees, 21 minutes) one second of longitude is m. Geographic coordinates appear on all standard topographic maps and, on some maps, they are the only method of locating and referencing a point. Universal Transverse Mercator Grid In Canada, the transverse mercator projection is used for most topographic mapping. The Universal Transverse Mercator Grid (UTM grid) is designed to cover the world between 84 degrees north and 80 degrees south latitude. As there is no land north of 84 degrees, the whole of Canada is covered by the UTM grid. The grid is used for point referencing in the same manner as the geographical coordinates but the grid is printed right on the map. To keep distortion due to the spherical shape of the earth to a minimum, the surface of the globe is projected down to a flat surface in sixty north-south strips, called zones. Each zone is six degrees of longitude wide. The UTM grid is placed over each zone. A grid normally has the following spacing between lines: 1:5,000 and larger scales 100 metre squares 1:5,000 to 1:100,000 1 kilometre squares smaller than 1:100, kilometre squares The grid lines are numbered eastward and northward from the southwest corner. The grid lines on the map running north-south are called easting lines or x coordinates because they are used to measure the distance of points eastward from the left side of the grid. The grid lines running

27 Chapter 9: Mathematics of Graphing and Surveying 9.27 east-west are called northing lines or y coordinates because they allow the measurement of the distance that point lie to the north of the equator. The actual measurements are called the eastings and northings or x and y coordinates of a point. Eastings are always given before northings. The UTM zones covering Canada are number 7 to 22 and shown in Figure FIGURE 9.20 UTM Zones and Central Meridians for Canada National Topographic System The National Topographic System (NTS) was designed in 1925 by the Ministry of Energy, Mines and Resources (then referred to as the Department of the Interior). Because these were the days of imperial units of measure, Canada was provided with a series of map scales at 1, 2, 4, 8, and 16 miles to the inch. On the NTS grid, the north-south lines are 8 degrees apart and the east-west lines are 4 degrees apart. The lines formed the boundaries of the maps scaled at 16 miles to the inch. As illustrated in Figure 9.21, the numbering is methodical: the numbers in the columns increase toward the north and the numbers increase by 10 in the east-west rows. FIGURE 9.21 The Primary Quadrangles of the NTS System

28 9.28 Foundations of Real Estate Mathematics FIGURE 9.22 The Breakdown of the Primary Quadrangle In the next larger series, 8 miles to the inch, each sheet covers one quarter of the area of the 16 mile sheet. The numbering of the 8 mile series starts with the basic quadrangle number: NW, NE, SW, and SE. The 4 mile series divides the 8 mile series into 16 divisions, which are numbered A to P. Numbering of the 4 miles series begins in the SE corner and proceeds west. The 2 miles series divides each quarter of the 4 miles series into NW, NE, SW, and SE quadrangles. Finally, the 1 miles series divides the 4 mile series into 16 divisions, numbered 1 to 16 beginning in the SE corner. Thus, a 1 mile map could be referenced as 31 A/9. The mile series are now converted to metric equivalents as follows: 1 mile 1:50,000 series 2 mile 1:125,000 series 4 mile 1:250,000 series 8 mile 1:500,000 series 16 mile 1:1,000,000 series In 1953, the 1:25,000 series was added to NTS. Each 1:50,000 sheet was divided into 8 smaller quadrangles numbered A to H starting in the SE corner. The 1:1,000,000 series no longer exists. SUMMARY This chapter outlined graphing techniques, the measurement of shapes, and the basics of surveying. The intent was to give students a basic understanding of approaches to describing and measuring both land and buildings. Most buildings are rectangular and therefore basic geometry is all that is needed. For more complicated and irregular shapes, geometric and trigonometric functions may be necessary. However, in practice, registered plans will often suffice for the measurements required for a real estate professional s work.

29 Chapter 9: Mathematics of Graphing and Surveying 9.29 APPENDIX 9.1: EXAMPLES OF LIVING AREA AND CALCULATIONS Example 7: Two and One-half Storey without Dormers Exterior length times width multiplied by two (Area A and B), plus length times width to knee wall (angle where wall meets ceiling; ceiling height must be greater than 5 feet; at least 50% of space must have height greater than 7 feet) and again account for width of exterior walls. Finished basement has same dimensions as Area A. Consider full second, third, etc., levels where ceiling height is acceptable. ABOVE GRADE FINISHED FLOOR AREA: Area A: x 7.65 = 78.8m 2 Area B: x 7.65 = 78.8m 2 Area C: x 3.95 = 40.7m 2 Total = 198.3m 2 BELOW GRADE FINISHED FLOOR AREA: Basement: x 7.65 = 78.8m 2 Total = 78.8m 2 AREA C BASEMENT & B & BASEMENT MAIN FLR., 2nd FLR. & BELOW GRADE FINISHED FLR AREA C 3rd FLR Example 8: Bi-Levels, Raised Bungalows or Split Entries Two elevation drawings are shown. The first drawing is without an overhang and the second drawing includes an overhang. The calculations assume the former; finished basement has same dimensions as Area A. Basement in Bi-levels are usually fully developed and quite often finished below grade floor area. ABOVE GRADE FINISHED FLOOR AREA: Area A: x = 109.2m 2 Total = 109.2m 2 BELOW GRADE FINISHED FLOOR AREA: Basement: x = 109.2m 2 Total = 109.2m & BASEMENT BASEMENT BASEMENT Example 9: Split Level or Three Level Side Split Consider Areas A and B above grade. Area under right side of diagram is a crawl space, approximately 60 cm to 1.2 m (two to four feet) in height. Finished basement has same dimensions as Area B and crawlspace has same dimensions as Area A. ABOVE GRADE FINISHED FLOOR AREA: Area A: 7.25 x 8.18 = 59.3m 2 Area B: 5.20 x 8.97 = 46.6m 2 Total = 105.9m 2 BELOW GRADE FINISHED FLOOR AREA: Basement: 5.20 x 8.97 = 46.6m 2 Total = 46.6m & BASEMENT UPPER FLR. & BELOW GRADE FINISHED FLOOR AREA BASEMENT & CRAWL SPACE MAIN FLR. & BELOW GRADE UNFINISHED FLOOR AREA CRAWL SPACE 8.18 BELOW GRADE UNFINISHED FLOOR AREA: Crawl Space: 7.25 x 8.18 = 59.3m 2 Total = 59.3m