# Solutions: Perimeter & Area of Polygons

#### Perimeter of Polygons

Exercise |
Problem |
Solution |

1 | Find the perimeter of a triangle the sides of which are 10 in, 14 in and 15 in. | P = 10 in + 14 in + 15 in = 39 in |

2 | A rectangle has a length of 12 cm and a width of 4 cm. Find its perimeter. | P = 12 cm + 12 cm + 4 cm + 4 cm = 32 cm |

3 | Find the perimeter of a regular hexagon with each side measuring 8 m. | P = 6(8 m) = 48 m |

4 | The perimeter of a square is 20 ft. How long is one side? | 20 ft ÷ 4 = 5 ft |

5 | The perimeter of a regular pentagon is 100 cm. How long is one side? | 100 cm ÷ 5 = 20 cm |

#### Area of Squares and Rectangles

Exercise |
Problem |
Solution |

1 | Find the area of a square with each side measuring 6 inches. | A = (6 in) (6 in) = 36 in^{2} |

2 | A rectangular yard has a length of 12 m and a width of 4 m. Find its area. | A = (12 m) (4 m) = 48 m^{2} |

3 | The area of a square is 25 square centimeters. How long is one side? | (x cm) (x cm) = 25 cm^{2}; x = 5 cm |

4 | A rectangular rug has an area of 28 square feet and a length of 7 ft. What is its width? | 28 ft^{2} ÷ 7 ft = 4 ft |

5 | A piece of paper has an area of 45 square inches and a width of 5 in. What is its length? | 45 in^{2} ÷ 5 in = 9 in |

#### Area of Parallelograms

Exercise |
Problem |
Solution |

1 | Find the area of a parallelogram with a base of 8 ft and a height of 3 ft. | A = (8 ft) (3 ft) = 24 ft^{2} |

2 | Find the area of a parallelogram with a base of 4 m and a height of 9 m. | A = (4 m) (9 m) = 36 m^{2} |

3 | The area of a parallelogram is 64 square inches and its height is 16 inches. Find its base. | 64 in^{2} ÷ 16 in = 4 in |

4 | A parallelogram has an area of 54 square centimeters and a base of 6 cm. Find its height. | 54 cm^{2} ÷ 6 cm = 9 cm |

5 | A parallelogram-shaped garden has an area of 42 square yards and a height of 6 yd. Find its base. | 42 yd^{2} ÷ 6 yd = 7 yd |

#### Area of Triangles

Exercise |
Problem |
Solution |

1 | Find the area of a triangle with a base of 16 ft and a height of 3 ft. | A = 1/2 (16 ft) (3 ft) = 24 ft^{2} |

2 | Find the area of a triangle with a base of 4 m and a height of 14 m. | A = 1/2 (4 m) (14 m) = 28 m^{2} |

3 | Find the area of a triangle with a base of 18 in and a height of 2 in. | A = 1/2 (18 in) (2 in) = 18 in^{2} |

4 | A triangular-shaped garden has an area of 36 square centimeters and a base of 6 cm. Find its height. (Hint: work backwards) | 36 cm^{2} = 1/2 (6 cm) (x cm); x = 12 cm |

5 | The area of a triangular-shaped rug is 12 square yards and its height is 3 yards. Find its base. (Hint: work backwards) | 12 yd^{2} = 1/2 (x yd) (3 yd); x = 8 yd |

#### Area of Trapezoids

Exercise |
Problem |
Solution |

1 | Find the area of a trapezoid with 5 ft and 7 ft bases, and a height of 3 ft. | A = 1/2 (5 ft + 7 ft) (3 ft) = 18 ft^{2} |

2 | Find the area of a trapezoid with 12 m and 6 m bases, and a height of 5 m. | A = 1/2 (12 m + 6 m) (5 m) = 45 m^{2} |

3 | Find the area of a trapezoid with 17 cm and 19 cm bases, and a height of 2 cm. | A = 1/2 (17 cm + 19 cm) (2 cm) = 36 cm^{2} |

4 | The area of a trapezoid is 48 square inches and its bases are 4 in and 8 in. Find its height. | 48 in^{2} = 1/2 (4 in + 8 in) (x in); x = 8 in |

5 | The area of a trapezoid is 77 square centimeters and its bases are 9 cm and 13 cm. Find its height. | 77 cm^{2} = 1/2 (9 cm + 13 cm) (x in); x = 7 cm |

#### Practice Exercises

Exercise |
Problem |
Solution |

1 | Find the perimeter of a triangle the sides of which have lengths 6 cm, 8 cm and 9 cm. | P = 6 cm + 8 cm + 9 cm = 23 cm |

2 | A rectangle has a length of 9 in and a width of 4 in. Find its perimeter. | P = 2 (9 in) + 2 (4 in) = 18 in + 8 in = 26 in OR P = 9 in + 9 in + 4 in + 4 in = 26 in |

3 | Find the perimeter of a regular octagon, the sides of which are 7 m long. | P = 8 (7 m) = 56 m |

4 | Find the area of a square the sides of which are 5 in long. | A = (5 in) (5 in) = 25 in^{2} |

5 | A rectangular yard has a length of 16 m and a width of 3 m. Find its area. | A = (16 m) (3 m) = 48 m^{2} |

6 | A rectangle has an area of 56 square feet and a width of 4 ft. What is its length? | 56 ft^{2} ÷ 4 ft = 14 ft |

7 | Find the area of a parallelogram with a base of 9 cm and a height of 5 cm. | A = (9 cm) (5 cm) = 45 cm^{2} |

8 | The area of a parallelogram is 24 square inches and its base is 4 in. Find its height. | 24 in^{2} ÷ 4 in = 6 in |

9 | Find the area of a triangle with a base of 7 m and a height of 4 m. | (7 m) (4 m) ÷ 2 = 14 m^{2} |

10 | Find the area of a trapezoid with 9 in and 13 in bases, and a height of 4 in. | A = [(9 in +13 in) ÷ 2] (4 in) = (11 in) (4 in) = 44 in^{2} |

#### Challenge Exercises

Exercise |
Problem |
Solution |

1 | A piece of paper has a length of 16 in and an area of 48 square inches. Find its width. | 48 in^{2} = (16 in) (x in); x = 3 in |

2 | The perimeter of a square-shaped rug is 36 meters. Find its area. | 36 m^{2} = (4) (x m); x = 9 m; If one side = 9 m, then Area = (9 m) (9 m) = 81 m^{2} |

3 | The area of a triangular-shaped garden is 72 square feet and its height is 24 ft. Find its base. | 72 ft^{2} = 1/2 (x ft) (24 ft); x = 6 ft |

4 | The height of a parallelogram is 6 more units than its base. What is the area of the parallelogram if its base is 15 cm? | b = 15 cm; h = 15 + 6 = 21 cm; A = (15 cm) (21 cm) = 315 cm^{2} |

5 | A dining room table has 8 sides of equal length. If one side is 15 in, what is the perimeter of the table? | P = (8) (15 in) = 120 in |

6 | A man has 30 ft of fence for a rectangular garden. What is the area of the largest garden he can build with whole-number sides? | P = 30 ft = 1 ft + 1 ft + 14 ft + 14 ft; A = 1 ft x 14 ft = 14 ft ^{2}P = 30 ft = 2 ft + 2 ft + 13 ft + 13 ft; A = 2 ft x 13 ft = 26 ft ^{2}P = 30 ft = 3 ft + 3 ft + 12 ft + 12 ft; A = 3 ft x 12 ft = 36 ft ^{2}P = 30 ft = 4 ft + 4 ft + 11 ft + 11 ft; A = 4 ft x 11 ft = 44 ft ^{2}P = 30 ft = 5 ft + 5 ft + 10 ft + 10 ft; A = 5 ft x 10 ft = 50 ft ^{2}P = 30 ft = 6 ft + 6 ft + 9 ft + 9 ft; A = 6 ft x 9 ft = 54 ft ^{2}P = 30 ft = 7 ft + 7 ft + 8 ft + 8 ft; A = 7 ft x 8 ft = 56 ft ^{2}The largest garden he can build has an area of 56 ft ^{2} |

7 | A cardboard cut-out has 5 sides of equal length. If its perimeter is 65 cm, what is the length of one side? | P = 65 cm = 5 (x cm); x = 13 cm |

8 | A chess board has an area of 49 square inches. Find its perimeter. | A = 49 in^{2} = (x in) (x in); x = 7 inP = 4 (7 in) = 28 in |

9 | Room carpet costs $25 per square yard. If the area of a bedroom is 20 square yards, how much will the room carpet cost? (No decimal point) | ($25)(20) = $500 |

10 | A mirrored plate has 6 sides of equal length. If its perimeter is 72 cm, what is the length of one side? | P = 72 cm; 72 cm ÷ 6 = 12 cm |