Center for Academic Excellence. Area and Perimeter

Transcription

1 Center for Academic Excellence Area and Perimeter Tere are many formulas for finding te area and perimeter of common geometric figures. Te figures in question are two-dimensional figures; i.e., in some sense tey ave lengt and widt. A line, in contrast, as only lengt (one dimension). First we will consider a rectangle: a four-sided figure wit four rigt angles and consequently a memer of te class of 4-sided figures called parallelograms. Recall te definition of a parallelogram: a 4-sided figure wit opposite sides parallel and equal. A B AB DC AD BC AB = DC AD = BC AD = BC D C Glossary: means parallel or is parallel to AB means te measure (lengt) of AB Of course, sometimes we are a it casual and say simply AB= DC and mean te same ting, teir lengts are equal. Te marks on te figure indicate equality. A B Te mark,, on AB and DC means tey are equal in lengt; Similarly,, indicates tat AD = BC. AD = BC D C We started y remarking tat tere are many formulas for finding te area (we ll consider perimeter later) of two-dimensional ( also called plane) figures. E F Look at te rectangle wit lengt of 3 units (inces, feet, centimeters, etc.) and widt of 2. Te square (a rectangle wit all 4 sides equal), Q is one unit (let s say 1 inc) on a side and tus is 1 square inc. You can count te numer of square w=2 inces in te figure EFGH. It is clearly 6. Wen you look at its dimensions, lengt, l = 3 inces, widt, w = 2 inces, can you see ow to find te area in a quicker manner tan counting te square inces? Of course, you multiply te lengt y te widt; H l=3 G in symolic notation: A=lw. In tis case, A = 3(2) = 6 sq. in. Any letters may e used to represent te dimensions; for example, we migt use and, wic conveniently are te initials of ase and eigt wic are commonly used terms in geometry. Center for Academic Excellence, University of Saint Josep, Revised

2 Te preceding discussion was intended to suggest tat you do not ave to memorize te formula for te area of a rectangle; rater, wit a quick sketc, you can figure it out for yourself. Now, let s use te formula, A =, to derive some oter formulas. Take a rectangular piece of paper, and cut it (or imagine cutting it) as sown: Now you ave a triangle (actually two). Wat is te area of te saded one? Clearly we cut te paper in alf so if te entire (rectangular) seet ad an area, A=, wat is te area of te triangle? A= 1 Do you ave to memorize it wen it s tat easy to figure out? Now take anoter rectangular seet and cut along te dotted line (in imagination if you prefer). Te result, if you move te little triangle, looks like te figure on te rigt. Wat is te figure on te rigt? A parallelogram. Wat is its area? Te same as it was efore you moved te triangular piece. So, wat is te formula for te area of a parallelogram? A= Rectangle A= Summary Parallelogram A= Center for Academic Excellence, University of Saint Josep, Revised

3 Triangle A=1/2 Wat aout te area of a circle: A circle is two-dimensional also and you proaly rememer te formula for its area, A=Πr2. Is it two- dimensional? Only one dimension, r, te radius, is given; ut, notice tis re-writing: A=Πr r were te dimension r is used twice. r is multiplied y r just as is multiplied y in oter area formulas. You may skip te next paragrap if you wis. It s just a demonstration of te appropriateness in te formula of Π, wic we may approximate, Π=3.14. Consider a square wit an inscried circle. Divide te large square into 4 equal smaller squares. Notice tat te lengt of te side of a small square is r, te radius of te inscried circle. Wat is te area of one small square? r r or r2. Wat is, terefore, te area of te large square? 4r2. Te corners of te large square outside te circle are saded to call attention to te fact tat te area of te circle is smaller tan te area of te large square. Te area of te large square is 4r2; te area of te circle is 3.14 r2. Tis seems reasonale. (Notice tat we just slipped te area of a square formula A=r2 into te discussion. After all, a square is a rectangle wit A=. But in our example ot te ase and eigt were r so A=r2 was OK. If we called te lengt of te side of te square s we would say A=s2. Perimeter Perimeter is te measurement around someting. Te perimeter of a rectangle of lengt l and widt w could simply e found y adding l+w+l+w. Just let your finger start at Q and walk around te figure. Of course, Perimeter, P=l+w+l+w could e written and calculated more efficiently y writing P=2l+2w. A little algera would give yet anoter form. Factor out te 2 in 2l+2w and te formula ecomes P=2(l+w). Center for Academic Excellence, University of Saint Josep, Revised

4 Since you know tat perimeter means te measure around, you don t really need a formula; just make sure you add up all te lengts tat make up te outside of te figure. w Q l Example: Wat is te perimeter of te figure ABCDEFGH? A 4 B 3 D 4 F 7 H 11 G If you start at A and add =44, you see tat te perimeter is 44 units. If it represents a figure measured in inces, 4 F te perimeter is 44 inces. Te perimeter of a circle is called te circumference and te formula for circumference is C=2Πr or since te diameter, d = 2r, C=Πd and tat s just a fact. Te way to sow it is a fact is to measure lots of circles (teir circumference and diameter) and sow tat te circumference divided y te diameter equals approximately 3.14 or Π. In symols, C = Π. Certainly, it is efficient to ave a list of formulas to find area and perimeter ut if te formulas are just unces of letters tat don t ave meaning it is muc too easy to forget tem or mix tem up. Te goal of tis paper is to convince you tat you can use your common sense and experience to recover forgotten formulas. Or, peraps it s to sow you tat you can often find area or perimeter witout using a formula. Practice 1. To estimate ow muc fertilizer you need to use on a garden would you find te area or te perimeter of te garden? 2. To figure ow muc fencing you need for your garden would you find te area or te perimeter of te garden? For exercises 3-6 find (a) te area and () te perimeter. 3. A rectangle, lengt = 3 feet; widt = 1 7/8 feet 4. A rectangle, lengt = 17 in.; widt = 11 in. Center for Academic Excellence, University of Saint Josep, Revised

5 5. Te triangle sown (measured in inces) Te parallelogram sown (measured in centimeters) 5 10 Find (a) te area and () te circumference of te circles wit dimensions given in exercises r = 3 in. 8. d = 11 ft 9. r = 4.5 cm 10. Find (a) te area and () te perimeter of te figure sown. 10in. 4 in. ANSWERS 1. area 6. (a) 40 sq. cm () 30 cm 2. perimeter 7. (a) sq. in. () in. 3. (a) 5 5/8 sq. ft () 9 ¾ ft 8. (a) sq. ft () ft 4. (a) 187 sq. in. () 56 in. 9. (a) sq. cm () cm 5. (a) 15.6 sq. in. () in. 10. (a) sq. in. () in. Center for Academic Excellence, University of Saint Josep, Revised