1. An NFL playing field (not counting the end zones) is 300 feet long and 160 feet wide. What is the perimeter? What is the area?

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1 Geometry: Perimeter and Area Practice 24 Many sports require a rectangular field of play which is a specific length and width. Use the information given in the problems below to compute the perimeter and area of each field of play. Reminder The perimeter of a rectangle is computed by adding the length and width and multiplying by 2. The area of a rectangle is computed by multiplying the length times the width. Remember: P = (l + w) x 2 and A = l x w 1. An NFL playing field (not counting the end zones) is 300 feet long and 160 feet wide. What is the perimeter? What is the area? 2. An NBA basketball court is 94 feet long and 50 feet wide. What is the perimeter? What is the area? 3. A major league baseball diamond is a square 90 feet long on each side. What is the perimeter? What is the area? 4. An ice hockey rink is 100 feet wide and 200 feet long. What is the perimeter? What is the area? 5. A field hockey playing area is 100 yards long and 60 yards wide. What is the perimeter? What is the area? 6. A softball diamond is a square 65 feet long on each side. What is the perimeter? What is the area? 7. A soccer field is 73 meters wide and 100 meters long. What is the perimeter? What is the area? 8. The playing area of a Canadian football field (not counting the end zones) is 110 yards long and 65 yards wide. What is the perimeter? What is the area? 27

2 Geometry: Area Practice 25 Lawn Magic is a business run by three sixth grade friends who earn money mowing their neighbors lawns. They charge by the square foot so they need to know the area of each lawn they mow. Help Lawn Magic compute the area in square feet of each lawn described below. Formulas to Remember Area of a rectangle = base times height (or length times width) Area of a parallelogram = base times height Area of a triangle = base times height divided by Lawn Magic did your neighbor s lawn which is a rectangular shape 12 feet high and 20 feet long at the base. What is the area? square feet 2. Lawn Magic mowed Mr. Crick s parallelogram-shaped lawn which has a height of 15 feet and a base of 30 feet What is the area? square feet 3. Mr. Ford s lawn is a parallelogram with a height of 23 feet and a base of 45 feet What is the area Lawn Magic will mow? square feet 4. Mrs. Jopp s lawn is triangular with a height of 12 feet and a base of 40 feet What is the area that Lawn Magic will mow? square feet 5. Lawn Magic mowed Mr. Lee s front lawn which is a rectangle 43 feet high and 97 feet at the base. What is the area they mowed? square feet 6. Mr. Dapper s back lawn is a triangle with a height of 33 feet and a base of 70 feet What is the area? square feet 7. Mrs. Smith s side lawn is a parallelogram 22.4 feet high and 30 feet at the base. What is the area Lawn Magic will mow? square feet 8. Lawn Magic mowed Ms. Brown s front lawn, a triangle 12.5 feet high and 14 feet at the base. What is the area they mowed? square feet 9. What is the area of a triangular lawn 16.6 feet high and 12 feet at the base? square feet 10. What is the area of a square lawn 22 feet on each side? square feet 28

3 Answer Key (cont.) 3. no 4. 5 m.p.h m.p.h. 6. the scale doesn t go 0 to start at 0/use a different scale thousand dollars 11. the scale is distorted, starts at thousand dollars 13. scale starts at 40 thousand dollars 14. starts at 0 and go to 70 Page feet 48,000 feet feet 4,700 feet feet 8,100 feet feet 20,000 feet yd. 6,000 yd feet 4,225 feet m 7,300 m yd. 7,150 yd. 2 Page feet feet ,035 feet feet ,171 feet ,155 feet feet feet feet feet 2 Page C = πd C = 3.14 x centimeters 2. C = πd C = 3.14 x centimeters 3. C = 2πr C = 2 x 3.14 x centimeters 4. C = πd C = 3.14 x centimeters 5. C = πd C = 3.14 x centimeters 6. C = 2πr C = 2 x 3.14 x inches 7. C = 2πr C = 2 x 3.14 x inches 8. C = 2πr C = 2 x 3.14 x centimeters Page A = πr 2 A = 3 x 3 x cm 2 2. A = πr 2 A = 3.14 x 8 x inches 2 3. A = πr 2 A = 3.14 x 6 x cm 2 4. A = πr 2 A = 3.14 x 7 x millimeters 2 5. A = πr 2 A = 3.14 x 9 x millimeters 2 6. A = πr 2 A = 3.14 x 2 x feet 2 7. A = πr 2 A = 3.14 x 4 x feet 2 8. A = πr 2 A = 3.14 x 4.5 x cm 2 9. A = πr 2 A = 3.14 x 3.5 x cm A = πr 2 A = 3.14 x 1.15 x cm 2 Page inches cm inches inches inches cubic puzzles cubic magnifying glasses 8. 1,000 cm 3 blocks games 10. 1,728 cubic puzzles Page library 2. town hall 3. gas station 4. (-11, 1) 5. (4, -4) 6. (-5, -9) 7. park 8. (-10, -7) 9. (-9, 5) 10. general store 11. drug store 12. III 13. I 14. II Page / / / /3 3. 9/ / / /5 5. 1/ /27 Page n = n = n = 41 n = n 29 = 61 n = n = 53 n = n = 43 n = n/4 = 12 n = n x 12 = 96 n = 8 8. n/8 = 11 n = n x 19 = 190 n = /n = 6 n = 7 Page :4 or 5/4 2. 4:5 or 4/5 3. 2:5 or 2/5 4. 5:2 or 5/2 5. 3:5 or 3/5 6. 5:3 or 5/3 7. 4:3 or 4/3 8. 3:4 or 3/4 9. 2:3 or 2/ :2 or 3/ :5 or 7/ :7 or 5/ :7 or 3/ :3 or 7/ :2 or 12/2 or 6:1 or 6/ :12 or 2/12 or 1:6 or 1/ :7 or 3/ :3 or 7/3 Page :4 :: 20:n n = 80 feet 2. 1:2 :: 25:n n = 50 feet 3. 3:15 :: 9:n n = 45 m 4. 4:1 :: 100:n n = 25 stories 5. 3:10 :: 33:n n = 110 yd. 6. 3:10 :: 15:n n = 50 m 7. 5:3 :: n:30 n = 50 inches 8. 7:2 :: 42:n or 2:7 :: n:42 n = 12 inches Page (58.67) (82.8) 3. 1, (89.2) (13.8) (5.25) (13.46) (6.8) Page (46, 47, 48, 49, 50, 52, 52, 52, 53, 54, 56) (47, 49, 55, 56, 57, 58, 59, 59, 59, 60, 60, 61, 63) (57, 59, 59, 60, 61, 61, 63, 63, 65, 66) 59, 61, (47, 49, 49, 49, 51, 52, 53, 54, 55, 57, 59) (39, 40, 44, 44, 45, 48, 50, 55, 57, 57, 58, 60, 60, 61) 44, 57, Page C 6. C 2. D 7. B 3. B 8. D 4. A 9. B 5. A 10. D Page B 6. A 2. D 7. C 3. C 8. A 4. A 9. B 5. D 10. C Page A 6. B 2. B 7. D 3. C 8. C 4. B 9. A 5. D 10. D Page C 6. B 2. C 7. A 3. B 8. D 4. D 9. B 5. D 10. C Page C 6. A 2. C 7. C 3. A 8. B 4. B 9. D 5. D 10. C Page C 6. C 2. A 7. A 3. B 8. B 48

4 Unit 7: Polynomials Name Area of Shaded Region Directions: Find the area of each shaded region in simplest terms. 165

5 Appendix B: Answer Keys Guided Practice Book Answers (cont.) 67

6 Area 6.7 Name Date Directions: Use the formulas to calculate both the area and perimeter for each rectangle. Think about the best way to record each answer, whether inches, square inches, feet, square feet, yards, or square yards. Complete the table below ft in yds. 9 ft. 32 in in m 56 in. 19 yds. 44 m yds. 27 yds. 7. A rectangle has a long side of 7.4 in. and a short side of 4 in. Find its perimeter and area. 8. A square wall has sides measuring 19 yds. What is the perimeter and what area do the bricks cover? area perimeter 63

7 Answer Key Page m = 100 cm, 1 kg = 1,000 g, 1 cm = 10 mm 2. 1 km = 1,000 m, 1 ton = 1,000 kg, 1 L = 1,000 ml 3. 1 km = 100,000 cm, 1t = 1,000,000 g, 1 L = 1,000 cm 4. To convert cm to km, we divide by 1, To convert g to kg, we divide by 1, To convert ml to L, we divide by 1, km cm 9. 9,600 g m mm kg 13. 8,600 g tons 15. 9,210 kg ,240 ml m 18. 5,100 ml cups pints quarts gallon pounds pounds feet feet Student Pages Page 63 Area Perimeter sq. ft. 48 ft. 2. 1,024 sq. in. 128 in sq. yards 62 yards 4. 1,936 sq. m 176 m sq. in. 128 in. 6. 2,862 sq. yards 266 yards sq. in in sq. yards 76 yards Page 69 Answers will vary. Page ,368.1 cm in m yards ft yards m in. 2 Page 81 Part A units cubed 2. 9 units cubed units cubed units cubed 5. 1,443 mm cubed mm cubed in. cubed ft in in. cubed 120

8 2 Practice Computing Perimeters of Rectangles To compute the perimeter of a rectangle, add the length and the width and then multiply by cm cm = 15.8 cm 15.8 cm x 2 = 31.6 cm P = 31.6 cm 12.5 cm 3.3 cm Directions: Use the information on page 9 to compute the perimeters of these rectangles. Remember to label the unit of measurement inches, feet, yards, centimeters, meters in your answers cm cm 2.4 cm 3.3 cm P = P = cm ft cm 3 ft. P = P = in cm in cm P = P = Directions: Use a ruler and the information on pages 5 and 9 to help you measure and compute the perimeters of these rectangles. 7. a math book cover 8. a sheet of paper length length width width P = P = 9. a paperback book cover 10. a desk length length width width P = P = 10

9 2 Practice To compute the perimeter of a regular polygon, in which all sides are equal, multiply the length of one side by the number of sides. Computing Perimeters of Regular Polygons 4.9 m Directions: Compute the perimeter of each of the regular polygons illustrated below. Remember to label the unit of measurement inches, feet, yards, centimeters, meters in your answer. 4.9 m x 4 = 19.6 m 5.2 cm in. 6.1 m ft. 9.3 m yd cm m 11

10 2 Practice Computing Perimeters of Irregular Polygons and Circumferences of Circles To compute the perimeter of an irregular polygon, add the lengths of the sides. 4.7 m 2.8 m 3.5 m P = 4.7 m m m m = 17.3 m Directions: Use the information on pages 5 and 9 to help you compute the perimeters of these polygons. Remember to label the unit of measurement inches, feet, yards, centimeters, meters in your answer. 1. P = 4.3 m 8.4 m 4.1 m 2.9 m 7.3 m 6.9 m 7.8 m 6.3 m 2. P = 3. P = 5 in. 6 1 in P = 4 1 ft. 2 3 ft. 8 1 in. 2 6 in. 2 1 ft. 2 5 ft ft. Directions: Use the information on page 9 to help you compute the circumferences of these circles. (C = 2πr or C = πd) 5. r = 4 m C = 6. r = 6 in. C = d = 10 cm C = d = 7 m C = 12

11 3 How to Compute Area Facts to Know Area of a Rectangle The area of a rectangle is computed by multiplying the length times the width. The formula is written: A = l x w (Area = length times width) or A = b x h (Area = base times height). The answer is given in square units. They are usually abbreviated like this: 4 sq. m or 4 m 2. Example: This rectangle is 6 centimeters long and 2 centimeters wide. 2 cm 6 cm A = l x w A = 2 cm x 6 cm A = 12 cm 2 Area of a Parallelogram The area of a parallelogram is computed by multiplying the base times the height. The formula is written: A = b x h or Area = base times height. Example: This parallelogram has a base of 12 cm and a height of 3 cm. Area of a Triangle The area of a triangle is computed by multiplying 1 times the base 2 times the height. A triangle is always one half of a rectangle or parallelogram. b x h The formula is written: A = 1 (b x h) or 2 2 Example: This triangle has a base of 10 cm and a height of 4 cm. Area of a Circle The area of a circle is computed by multiplying pi (which is approximately 3.14) times the radius times itself. 3 cm 12 cm A = b x h A = 12 cm x 3 cm A = 36 cm 2 4 cm 10 cm A = 1 (b x h) 2 A = 1 x 10 cm x 4 cm 2 A = 20 cm 2 Example: This circle has a radius of 4 cm. r = 4 cm A = πr 2 (Area = pi x the radius x the radius) A = 3.14 x 4 cm x 4 cm A = cm 2 13

12 3 Practice Computing the Areas of Rectangles and Parallelograms All rectangles are also parallelograms. They have two pairs of parallel sides. w The formula for the area of a rectangle is A = l x w or A = b x h The formula for the area of a parallelogram is A = bh rectangle l parallelogram h b Directions: Use the information on page 13 to compute the areas of these rectangles and parallelograms. Remember to indicate the unit square feet, square meters, square inches, etc. with the answer m A = 18 yd. 7 yd. A = 3. 5 m m 9 cm A = 1.9 m A = 7.5 cm in ft. 5 in. A = 4 ft. A = mm m A = 100 mm A = 40 m 14

13 3 Practice Computing the Areas of Triangles This is the formula for computing the area of a triangle: A = 1 2 b x h (or) A = b x h 2 A = 1 2 x 8 x 4 A = 16 m 2 4 m Directions: Use the information on page 13 to compute the areas of these triangles. Remember to indicate the unit square feet, square meters, square inches, etc. with the answer. 8 m 1. A = 2. A = 6 ft. 9 yd. 8 ft. 10 yd. 3. A = 4. A = 5.3 cm 6.6 cm 4.4 cm 8.4 cm 5. A = 6. A = 27 in. 9.1 m 30 in m 7. A = 8. A = 11.3 cm 9.5 m 20.6 cm 18.2 m 15

14 Answer Key Page " " /4" /16" Answers will vary. Pages 7 and 8 Answers will vary. Page cm cm cm ft in cm Answers will vary. Page cm in m ft m yd cm m Page m m in ft m in cm m Page m yd cm m ft /4 in ,680 m ,500 mm 2 Page ft yd cm cm in m cm m 2 Page m cm cm cm ,256 cm ft in ,962.5 m 2 Page m ft cm in yd m m cm ,728 ft /8 ft. 3 Page m cm cm in ,240 cm ,538.6 ft. 3 Pages Answers will vary. Page lbs. 4 oz ton 300 lbs. 3. 4,000 cassettes pills ,000 pills 6. 2,000 dictionaries 7. 12,000 staplers people mg or 1/2 g kg 11. 4,400 kg 12. 2,200 clips 13. 6,400 calculators cameras Page fl. oz fl. oz fl. oz fl. oz fl. oz fl. oz fl. oz fl. oz fl. oz fl. oz qt qt fl. oz qt ,920 fl. oz fl. oz fl. oz fl. oz pints cups pints 22. 1,280 fl. oz cups fl. oz fl. oz. Page ml ml 3. 1,000 ml ml ml ml 7. 3,840 ml L L L L L L L Page qt ml ml ml pennies ml fl. oz ml quarters L times cups Page acute obtuse straight right acute obtuse reflex reflex right acute Page <BAC = <CBA = <ACB = ABC = <CDE = <ECD = <DEC = DEC = <LMN = <MNL = <MLN = LMN = <MNO = <OMN = <MON = MNO = <XYZ = <ZXY = <YZX = XYZ = <WPO = <POW = <PWO = WPO = 180

15 7 How to Understand Quadrangles Facts to Know A quadrilateral is the name given to a plane figure with four straight sides. In other words, it s a foursided polygon. Quadrilaterals are also called quadrangles. Quadrangles have four straight sides and four angles. Kinds of Quadrangles A 4'' C Square A square has four sides, all the same length. It has four right angles and two pairs of parallel sides. AB CD and AC BD 4'' B 4'' 4'' D E 7'' F Rectangle A rectangle has equal opposite sides. It has four right angles and two pairs of parallel sides. EF GH and EG FH 4'' G 7'' B 4'' H Rhombus A rhombus has four equal sides like a square. It has two pairs of parallel sides. It has equal opposite angles. AB CD and AC BD A 2'' 2'' 2'' 2'' D Parallelogram A parallelogram has equal opposite sides like a rectangle. It has two pairs of parallel sides. It has equal opposite angles. A rhombus is a kind of parallelogram. MO NP and MN OP N 2'' M 4'' C 4'' P 2'' O Trapezoid A trapezoid has only one pair of parallel sides. AB CD 30 C 4'' A 2'' 6'' B 5'' D

16 7 How to Understand Quadrangles Facts to Know (cont.) Finding Perimeter Perimeter is the distance around a figure. To find the perimeter of a triangle, a quadrilateral, or a polygon a figure with three sides or more you add the lengths of the sides. 4' 5' 4' + 5' + 7' = 16' The perimeter of the triangle is 16'. 7' 5'' 6'' + 5'' + 4'' + 9'' = 24'' The perimeter of this trapezoid is 24''. 6'' 4'' 3 ft. 9'' 3 ft. 3 ft. 3 ft. + 3 ft. + 3 ft. + 3 ft. = 12 ft. The perimeter of this square is 12 ft. 3 ft. There are formulas for finding perimeter, but sometimes just adding up the sides is faster. For the perimeter of a square: P (Perimeter) = 4s (side). For the square above, enter the numbers in the formula, P = 4(3 ft.) or 12 ft. For the perimeter of a rectangle or parallelogram: P (Perimeter) = 2w (width) + 21 (length). Let s say you knew the lengths of two sides of a rectangle. 5 ft. You could use the formula: P = 2(5 ft.) + 2(14 ft.), which is 38 ft. 14 ft. 31

17 7 Practice Directions: Read the clues and answer the questions. Identifying and Finding the Perimeter of Quadrangles 1. It has 2 pairs of equal opposite sides like a rectangle. It has two pairs of parallel sides. It has 2 pairs of equal opposite angles. What kind of quadrangle is it? 2. It has one pair parallel sides. What kind of quadrangle is it? Directions: Identify the following quadrangles. 5' '' 5' 5' 5'' 5'' ' 8'' '' 8'' 3'' '' '' 6'' '' 50 6''

18 7 Practice Directions: Find the perimeter. Identifying and Finding the Perimeter of Quadrangles 30 ft ft. 11 ft. 9. A parallelogram has a perimeter of 13 feet. Its length is 3.75 feet. What is its width? 10. A square has a perimeter of 28 feet. What is the length of one side? _ 11. Grandma s garden has sides of 15 ft., 10 ft., 8 ft., and 12 ft. How much fencing does she need to keep the rabbits out? 12. Myra s poster of the Rockin Jellybeans measures 4 ft. by 3 ft. How much frame does she need to go around it? 33

19 8 How to Find the Areas of Different Geometric Shapes Facts to Know Area is the amount of space on a flat surface. Area is measured in square units: square inches, square feet, square miles, etc. Think of a surface covered with square tiles. In the drawing below, each tile is 1 square inch. 19 in. 9 in. If you counted them all, you would find there are 171 square inches in this 19 in. by 9 in. rectangle. But there s a simpler way to arrive at the total number of square inches. Finding the Area of Rectangles In the rectangle above, there are 9 rows of 19 square inches. A shortcut is to multiply 9 in. x 19 in., which would give you 171 square inches. The formula for finding the area of a rectangle is A (area) = l (length) x w (width). Here s an example using the formula: 4 ft. A = l x w A = 4 ft. x 3 ft. A = 12 ft. 2 3 ft. So, the area is 12 sq. ft. Always remember to add square or sq. in the answer to an area problem. Finding the Area of Squares A square is a quadrilateral with four equal sides. The formula to find the area is even simpler: A (area) = s (side) x s (side) or A = s 2. Using the formula, 5 ft. A = s x s A = 5 ft. x 5 ft. A = 25 ft. 2 5 ft. So, the area of the square is 25 sq. ft. 34

20 8 How to Find the Areas of Different Geometric Shapes Facts to Know (cont.) Finding the Area of Triangles To find the area of a triangle, you need to know the base and the height. The base can be any side of the triangle. The height of a triangle is its altitude. altitude The altitude is a line that is perpendicular to the base and extends from the angle opposite the base. Perpendicular lines form right (90 ) angles. base To find the area, the height or altitude must be given so that you can put the numbers into the formula, A (area) = 1 b (base) x h (height) or A = 1 bh. 2 2 Here s an example using the formula: A = 1 10 ft. x 10 x 8 (base) 2 A = 1 (80) 2 A = 40 So, the area of the triangle is 40 sq. ft. 8 ft. (height) Finding the Area of Parallelograms You use the same logic for finding the area of a triangle to find the area of a parallelogram. To find the area, the height must be given in the problem. The formula is A (area) = b (base) x h (height) or A = bh. 5 ft. (height) 9 ft. (base) Using the formula, A = 9 x 5 A = 45 So, the area of the parallelogram is 45 sq. ft. 35

21 8 Practice Directions: Answer the questions. Finding the Areas of Different Geometric Shapes 1. The high jump pit is a rectangle 12 feet long and 10 feet wide. What is the area? 2. Ingrid is purchasing a rectangular rug that measures 6 ft. by 8 ft. How many square feet of floor space will it cover? 3. A baseball field has 20 yards between bases. The Izaac Walton Ball Club wants to buy a tarp that will cover the field at night. How big will the tarp have to be? 4. What is the area of the square below? What is the area of the rectangle below? 10.5 in. 22'' 8'' 5. Ingrid s first rug was too small. So, she bought a 8 ft. by 10 ft. rug instead. It s going to go in a 10 ft. by 12 ft. room. How much floor space won t be covered by the rug? 10 ft. 8 ft. 12 ft. 10 ft. room rug 6. Find the area of a rectangle that has a length of 6.5 ft. and a width of 2.25 ft. 36

22 8 Practice Directions: Find the area of each triangle. Finding the Areas of Different Geometric Shapes 7. 5' 5' 3' Area = 8' '' 6'' 3'' 2'' Area = 9. 3' 5' Area = 4' Directions: Find the area of each parallelogram ' Area = 9.5' ' Area = 7' '' Area = 18'' 37

23 Answer Key Pages 7 and 8 1. d 2. g 3. b 4. h 5. b 6. e 7. b 8. e 9. a 10. f 11. c 12. g 13. d 14. f Pages 12 and b 2. f 3. a 4. f 5. b 6. g 7. d 8. e 9. b 10. e 11. c 12. h Page g 6. f Pages 20 and radius 2. diameter 3. chord 4. circumference 5. 4 ft in ft in in ft miles yds ft in ft. 2 Page acute 2. equilateral 3. right 4. isosceles 5. obtuse 6. scalene 7. acute 8. isosceles 9. acute 10. scalene 11. acute 12. equilateral Page acute and scalene acute and equilateral 5. D = 55 F = c = 2.5'' 8. b = 12' Pages 32 and parallelogram 2. trapezoid 3. rhombus 4. rectangle 5. trapezoid 6. parallelogram ft ft ft ft ft ft. Pages 36 and ft ft yds in. 2, 176 in ft ft ft in ft ft ft in. 2 Pages 40 and in in ,154 in in ft ft , ft ft. 3 or ft. 3 Pages 42 and m; 9.5 m m; 9 m m; 81 m m m; 4.64 m cm m; m m ft ft quarts m ,560 ft ,840 yards acres 16. 3,780,000 pounds 17. A = 5,024 cm 2 C = 251 cm minutes 19. r = 50 cm A = 7,850 cm 2 C = 314 cm time = min 20. r = 30 cm A = 2,826 cm 2 C = cm time = min Pages 44 and cm 2 = 1,024 cm 2 2. P = 2(4s) = 16 cm 3. P = 4(4s) = 32 cm 4. A = 4(1 x w) = 16 cm 2 5. A = 16(1 x w) = 64 cm Let side of square A = 1 cm Let the side of square B = 4 cm Area square A = 1 cm Area square B = 16 cm The area of square B is 16 times greater than the area of square A. 8. Area of rectangle = 70 cm x 30 cm = 2,100 cm 2 2,100 cm cm 2 = 2,700 cm 2 30 x 2,700 cm 2 = 81,000 cm 2 of wood 9. Yes, they have the same area. Since you multiply the base and height, and these two parallelograms use the same numbers, so it doesn t matter which is the base and which is the height. 48

SIXTH GRADE MATHEMATICS CHAPTER 10 AREA AND PERIMETER TOPICS COVERED:

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