Area of Figures – Chapter Problems
Area of Rectangles
Classwork
1. Find the area of a rectangle that has a length of 4.6 in. and a width of 3.4 in.
2. Annabelle wants to place new carpeting in her dining room. Her dining room is a rectangle with a length of 30 feet and a width of 15 feet.
a. How much carpeting does she need to buy to cover her entire dining room?
b. If the carpeting costs $7.50 per square foot, how much will it cost to carpet Annabelle’s dining room?
3. The diagonal of a rectangle is 26 feet, and its width is 10 feet. Find the length of the rectangle and its area.
4. The diagonal of a rectangle is 50 feet, and its length is 10 more feet than its width. Find the rectangle’s length, width, and area.
PARCC-type Question
5. The population density is the amount of people living per square mile. If the town of Mathville is a rectangular town that has a length of 12 miles, a width of 6 miles, and a population of 3,982 people, what is the population density of the town? Round your answer to the nearest hundredth.
Area of Rectangles
Homework
6. Find the area of a rectangle that has a length of 8.5 cm and a width of 3.7 cm.
7. Rick wants to place new carpeting in his living room. His living room is a rectangle with a length of 40 feet and a width of 20 feet.
a. How much carpeting does he need to buy to cover his entire living room?
b. If the carpeting costs $8.75 per square foot, how much will it cost to carpet Rick’s living room?
8. The diagonal of a rectangle is 30 feet, and its width is 6 feet less than its length. Find the rectangle’s length, width, and area.
9. A rectangle has a perimeter of 200 feet and its length is 4 times its width. Find the rectangle’s length, width, and area.
PARCC-type Question
10. The population density is the amount of people living per square mile. If the town of Algebraville is a rectangular town that has a length of 13 miles, a width of 10 miles, and a population of 12,576 people, what is the population density of the town? Round your answer to the nearest hundredth.
Area of Triangles
Classwork
Find the area of the triangle for #11-14.
11. 12.
13. 14.
15. Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an altitude from vertex B perpendicular to the opposite side.
Area of Triangles
Homework
Find the area of the triangle for #16-19.
16. 17.
18. 19.
20. Derive the formula A = ½ ac sin(B) for the area of a triangle by drawing an altitude from vertex A perpendicular to the opposite side.
Law of Sines
Classwork
Solve the triangle. Round your answers to the nearest hundredth.
21.
m∠A=__________, m∠B= __________, AC = __________
22.
m∠E=__________, m∠F= __________, DF = __________
23.
x = __________, y = __________, z = __________
24.
x = __________, y = __________, z = __________
Law of Sines
Homework
Solve the triangle. Round your answers to the nearest hundredth.
25.
m∠H=__________, m∠I= __________, GH = __________
26.
m∠J=__________, m∠L= __________, KL = __________
27.
x = __________, y = __________, z = __________
28.
x = __________, y = __________, z = __________
Area of Parallelograms
Classwork
29. Find the area of a parallelogram that has a base of 3.6 inches and a height of 8.2 inches.
30. A parallelogram has a height of 5.2 cm and an area of 48.88 cm2. Find the length of its base.
31. A parallelogram has a base length of 9.3 inches and an area of 56.73 in2. Find the height of the parallelogram.
32. A parallelogram has a base length of 7.5 cm. The length of its other side is 5 cm, and the obtuse angle between two of the sides is 133°. Find the height and area of the parallelogram.
33. A diagonal parking space creates a parallelogram. Its length is 21 feet. Its distance along the curb is 10.5 feet, and the acute angle that is made with the curb is 60°. Find the area of the parking space.
34. A window frame is in the shape of a parallelogram. Its base is 4 feet long and its other side length is 3 feet long. The acute angle between two of the sides is 48°. Find the height and the area of the window.
35. Mrs. Polygon is making a quilt for her granddaughter. The quilt is created by stitching together parallelograms that are different colors. Each parallelogram is 3 inches long, its other side is 2 inches long, and the obtuse angle between the two sides is 145°.
a. What is the area of each parallelogram used to make the quilt? Round your answer to the nearest hundredth.
b. The quilt requires 6 parallelograms horizontally and 12 parallelograms vertically. How much material will Mrs. Polygon need to make the quilt?
Area of Parallelograms
Homework
36. Find the area of a parallelogram that has a base of 2.5 feet and a height of 7.9 feet
37. A parallelogram has a height of 4.8 cm and an area of 32.16 cm2. Find the length of its base.
38. A parallelogram has a base length of 19.5 inches and an area of 232.05 in2. Find the height of the parallelogram.
39. A parallelogram has a base length of 8.3 cm. The length of its other side is 6.2 cm, and the acute angle between two of the sides is 48°. Find the height and area of the parallelogram.
40. A diagonal parking space creates a parallelogram. Its length is 19 feet 10 inches. Its distance along the curb is 12 feet 9 inches, and the acute angle that is made with the curb is 45°. Find the area of the parking space.
41. A window frame is in the shape of a parallelogram. Its base is 3.5 feet long and its other side length is 2.5 feet long. The obtuse angle between two of the sides is 127°. Find the height and the area of the window.
42. In the Polygon household, the living room floor is shaped like a parallelogram. The length of the room is 30 meters and the length of the side adjacent to the base is 20 meters. The obtuse angle between the two sides is 125°. Mrs. Polygon wants to install hardwood flooring in her living room.
a. What is the area of the living room?
b. If hardwood flooring costs $8.25 per square foot, how much will it cost to get the hardwood floor installed?
Area of Regular Polygons
Classwork
Calculate the perimeter and area of each regular polygon. Round your answers to the nearest hundredth.
43.
44.
45.
46.
Area of Regular Polygons
Homework
Calculate the perimeter and area of each regular polygon. Round your answers to the nearest hundredth.
47.
48.
49.
50.
Area of Circles & Sectors
Classwork
Find the area of the minor sector. Round to the nearest hundredth or leave your answer in terms of π.
51. 52. 53. 54.
Find the area of the major sector. Round to the nearest hundredth or leave your answer in terms of π.
55. 56. 57. 58.
Area of Circles & Sectors
Homework
Find the area of the minor sector. Round to the nearest hundredth or leave your answer in terms of π.
59. 60. 61. 62.
Find the area of the major sector. Round to the nearest hundredth or leave your answer in terms of π.
63. 64. 65. 66.
Area of Other Quadrilaterals
Classwork
Answer each question below.
67. A teacher has 15 desks in their classroom. Each desk was created using wood for the flat top surface and metal bars for the legs. The shape of all of the flat top surfaces is an isosceles trapezoid. The short and long bases have a length of 3 feet and 5 feet, respectively, and the acute angle formed by the legs and the long base is 68°.
a. Find the height of the trapezoid.
b. Find the total amount of wood required to make all 15 desks in the classroom.
68. A ruby was cut into the shape of a kite to make the ring shown to the right. If its diagonals measure 20 mm and 11 mm, what is the area of the stone?
69. A rhombic triacontahedron is a convex 3-D solid with 30 rhombic faces. Each rhombic face has acute angles that measure 63.43°. The length of the small diagonal is 2 inches.
a. Find the length of the long diagonal in the rhombus.
b. Find the area of one rhombic face.
c. Find the surface area (area of all of the faces) in this rhombic triacontahedron.
Find the area of each complex figure given below.
70. 71. The radius of each semicircle is 5 cm.
PARCC-type Questions
Solve the following word problems based on the information below.
The Bisect Building Company has created a building plan for the new patio for the Quadrilateral Family, shown in the figure.
72. The roof of the patio made from 2 isosceles trapezoids and 3 rectangles.
a. What is the area of the entire roof? Explain your answer.
b. Each bundle of shingles covers 36 ft2. Shingles cost $25.50 per bundle and must be purchased in full bundles. The builder has a budget of $125 for shingles. Did the Bisect Building Company budget enough money for the shingles? Explain your answer.
73. The patio will cover the Quadrilateral’s new hot tub (or Jacuzzi tub) and possibly a sidewalk that is 2 feet wide on all four sides. The hot tub is 6’ 10” x 6’ 10”.
a. How much concrete is needed to make the sidewalk surrounding the entire hot tub (or Jacuzzi tub)? Explain your answer.
b. If the price for sidewalk installation is $4.75 per square foot, how much will the Quadrilateral’s have to pay to get their sidewalk installed? Explain your answer.
c. Will the roof cover the hot tub (or Jacuzzi tub) and the sidewalk? Explain your answer.
Area of Other Quadrilaterals
Homework
Answer each question below.
74. A teacher has 12 desks in their classroom. Each desk was created using wood for the flat top surface and metal bars for the legs. The shape of all of the flat top surfaces is an isosceles trapezoid. The short and long bases have a length of 3.5 feet and 5.5 feet, respectively, and the acute angle formed by the legs and the long base is 57°.
a. Find the height of the trapezoid.
b. Find the total amount of wood required to make all 12 desks in the classroom.
75. The logo for Mitsubishi Motors is comprised of 3 congruent rhombi joined together at a center point to create a triangular figure. In each rhombus, the length of the long diagonal is 5 inches and the length of the short diagonal is 2.5 inches.
a. What is the area of each rhombus in the logo?
b. What is the total rhombus area of the logo?
76. A blue sapphire stone was cut into the shape of a kite to make a necklace as shown to the right. If its diagonals measure 2.2 cm and 1.1 cm, what is the area of the stone?
Find the area of each complex figure given below.
77. 78.
PARCC-type Questions
Solve the following word problems based on the information below.
Mrs. Skew is going to redesign her bedroom, shown in the picture to the right.
79. The first thing that she needs to do is replace her carpet. If carpet is sold for $6.75 per square foot, how much will it cost? Explain your answer.
80. Mrs. Skew is going to paint all of the walls, which are 8 feet high, with a 2 coats of paint. The room contains a doorway that is 3 ft by 7 ft, 3 windows measuring 4 ft by 3.5 ft, and a closet doorway that is 6 ft by 7 ft. The doorway, windows and closet doorway will not be painted.
a. What is the total amount of wall space needs to be covered with paint? Explain your answer.
b. If paint is sold in 1-gallon containers, and each gallon of paint covers 350 square feet and each can of paint costs $12.50, how much money does Mrs. Skew need to spend on paint? Explain your answer.
81. Given the figure below, explain why the area formula for a kite is A=12d1d2.
Area and Perimeter of Figures in the Coordinate Plane
Classwork
Calculate the perimeter & area of each figure below.
82. 83.
PARCC-type Questions:
84. The figure shows polygon ABCDEF in the coordinate plane with point A at (0, 3.71), point B at (1.64, 3.71), point C at (1.64, 3), point D at (2.89, 3), point E at (2.89, 0), and point F at the origin. Polygon ABCDEF can be used to approximate the size of the state of Utah with x and y scales representing hundreds of miles.
a. Based on the information given, how many miles is the perimeter of Utah?
b. At the end of 2010, the population of Utah was 2,763,885 people. Based on the information given, what was the population density at the end of 2010?
85. The town of Geometryville has Polygon Park on a plot of land. The figure represents a map of the Polygon Park showing the location of the Parking Area, Soccer Fields, and Playground. The coordinates represent points on a rectangular grid with the units in hundreds of feet.
a. There is a walking trail around the entire park, including the Parking area. What is the length of the walking trail? Express your answer to the nearest foot.
b. What is the area of the plot of land that does not include the parking area? Express your answer to the nearest square foot.
c. The town is planning to put a fence around the Playground and Soccer Fields. What is the total length of fencing needed? Express your answer to the nearest foot.