Composite Areas Quiz: Master Composite Areas Quiz

1) A playground is a half circle with diameter 12 m attached to a rectangle 12 m by 20 m. What is the approximate total area?

296.5 m²

353.7 m²

413.7 m²

500 m²

Area = rectangle + semicircle = (12 times 20) + ½ times π times 6² = 240 + ½ times 3.14159 times 36 = 240 + 56.55 = 296.55 ≈ 296.5 m². Option B gives 353.7, option C gives 413.7, option D gives 500, all of which overestimate the total.

Explanation

Area = rectangle + semicircle = (12 times 20) + ½ times π times 6² = 240 + ½ times 3.14159 times 36 = 240 + 56.55 = 296.55 ≈ 296.5 m². Option B gives 353.7, option C gives 413.7, option D gives 500, all of which overestimate the total.

2) A regular garden path 30 by 2 has a semicircle at each end with diameter 2. What is the approximate total area?

60 m²

63.1 m²

70 m²

73 m²

Area = rectangle + 2 semicircles = (30 times 2) + π times 1² = 60 + 3.14159 = 63.14 ≈ 63.1 m². Two semicircles with radius 1 equal one full circle. Option A gives 60, the rectangle alone with no semicircles added. Options C and D both overestimate the total.

Explanation

Area = rectangle + 2 semicircles = (30 times 2) + π times 1² = 60 + 3.14159 = 63.14 ≈ 63.1 m². Two semicircles with radius 1 equal one full circle. Option A gives 60, the rectangle alone with no semicircles added. Options C and D both overestimate the total.

3) A rectangle 12 by 4 is topped with a triangle with base 12 and height 6. What is the total area?

60 m²

72 m²

84 m²

96 m²

Area = rectangle + triangle = (12 times 4) + ½ times (12 times 6) = 48 + 36 = 84 m². Option A gives 60, option B gives 72, option D gives 96, none of which correctly apply both area formulas.

Explanation

Area = rectangle + triangle = (12 times 4) + ½ times (12 times 6) = 48 + 36 = 84 m². Option A gives 60, option B gives 72, option D gives 96, none of which correctly apply both area formulas.

4) A fountain base shaped like a rectangle 12 by 8 with a semicircle with diameter 12 attached has what total area?

120 m²

140 m²

152.5 m²

160 m²

Area = rectangle + semicircle = (12 times 8) + ½ times π times 6² = 96 + ½ times 3.14159 times 36 = 96 + 56.55 = 152.55 ≈ 152.5 m². Option A gives 120, option B gives 140, option D gives 160, none of which correctly compute both component areas.

Explanation

Area = rectangle + semicircle = (12 times 8) + ½ times π times 6² = 96 + ½ times 3.14159 times 36 = 96 + 56.55 = 152.55 ≈ 152.5 m². Option A gives 120, option B gives 140, option D gives 160, none of which correctly compute both component areas.

5) A garden path 12 by 3 has semicircles on both ends with diameter 3. What is the approximate total area?

43.1 m²

46 m²

56 m²

74 m²

Area = rectangle + 2 semicircles = (12 times 3) + π times (1.5)² = 36 + 3.14159 times 2.25 = 36 + 7.07 = 43.07 ≈ 43.1 m². Two semicircles equal one full circle with radius 1.5. Option B gives 46, option C gives 56, option D gives 74, all of which overestimate the total area.

Explanation

Area = rectangle + 2 semicircles = (12 times 3) + π times (1.5)² = 36 + 3.14159 times 2.25 = 36 + 7.07 = 43.07 ≈ 43.1 m². Two semicircles equal one full circle with radius 1.5. Option B gives 46, option C gives 56, option D gives 74, all of which overestimate the total area.

6) A square 8 by 8 has a 4 by 4 square shaded inside. What is the unshaded area?

44 cm²

48 cm²

52 cm²

60 cm²

Unshaded area = outer square minus inner square = (8 times 8) minus (4 times 4) = 64 minus 16 = 48 cm². Option A gives 44, option C gives 52, option D gives 60, none of which correctly subtract the inner square from the outer square.

Explanation

Unshaded area = outer square minus inner square = (8 times 8) minus (4 times 4) = 64 minus 16 = 48 cm². Option A gives 44, option C gives 52, option D gives 60, none of which correctly subtract the inner square from the outer square.

7) A 15 by 6 rectangle has a 5 by 6 rectangle shaded inside. What is the unshaded area?

60 m²

70 m²

80 m²

90 m²

Unshaded area = outer rectangle minus inner rectangle = (15 times 6) minus (5 times 6) = 90 minus 30 = 60 m². Option B gives 70, option C gives 80, option D gives 90 which is the full outer rectangle with nothing removed. Only 60 correctly subtracts the shaded inner region.

Explanation

Unshaded area = outer rectangle minus inner rectangle = (15 times 6) minus (5 times 6) = 90 minus 30 = 60 m². Option B gives 70, option C gives 80, option D gives 90 which is the full outer rectangle with nothing removed. Only 60 correctly subtracts the shaded inner region.

8) A running track shaped like a 60 by 20 rectangle with two semicircles of radius 10 m attached has what approximate total area?

1205.4 m²

1514.2 m²

1828.5 m²

1920 m²

Area = rectangle + 2 semicircles = (60 times 20) + π times 10² = 1200 + 3.14159 times 100 = 1200 + 314.16 = 1514.16 ≈ 1514.2 m². Two semicircles equal one full circle. Option A gives 1205.4, option C gives 1828.5, option D gives 1920. Only 1514.2 correctly adds the rectangle and full circle.

Explanation

Area = rectangle + 2 semicircles = (60 times 20) + π times 10² = 1200 + 3.14159 times 100 = 1200 + 314.16 = 1514.16 ≈ 1514.2 m². Two semicircles equal one full circle. Option A gives 1205.4, option C gives 1828.5, option D gives 1920. Only 1514.2 correctly adds the rectangle and full circle.

9) A rectangle 20 by 10 with a semicircle with diameter 20 attached on one side has what total area?

285.4 m²

314.2 m²

357.1 m²

400 m²

Area = rectangle + semicircle = (20 times 10) + ½ times π times 10² = 200 + ½ times 3.14159 times 100 = 200 + 157.08 = 357.08 ≈ 357.1 m². Option A gives 285.4, option B gives 314.2, option D gives 400, none of which correctly add both components.

Explanation

Area = rectangle + semicircle = (20 times 10) + ½ times π times 10² = 200 + ½ times 3.14159 times 100 = 200 + 157.08 = 357.08 ≈ 357.1 m². Option A gives 285.4, option B gives 314.2, option D gives 400, none of which correctly add both components.

10) A square 12 by 12 has a circle of radius 6 cut out. What is the approximate remaining area?

30.9 m²

44 m²

72.5 m²

113.1 m²

Area = square minus circle = (12 times 12) minus π times 6² = 144 minus 3.14159 times 36 = 144 minus 113.1 = 30.9 m². Option B gives 44, option C gives 72.5, option D gives 113.1 which is just the circle area alone. Only 30.9 correctly subtracts the circle from the square.

Explanation

Area = square minus circle = (12 times 12) minus π times 6² = 144 minus 3.14159 times 36 = 144 minus 113.1 = 30.9 m². Option B gives 44, option C gives 72.5, option D gives 113.1 which is just the circle area alone. Only 30.9 correctly subtracts the circle from the square.

11) A semicircle's area is always half that of a full circle with the same radius.

True

False

The answer is True. The area of a full circle is π times r squared. A semicircle is exactly half of a full circle, so its area is ½ times π times r squared. This relationship holds for any radius value. When solving composite area problems involving semicircles, always halve the full circle area formula.

Explanation

The answer is True. The area of a full circle is π times r squared. A semicircle is exactly half of a full circle, so its area is ½ times π times r squared. This relationship holds for any radius value. When solving composite area problems involving semicircles, always halve the full circle area formula.

12) A rectangle 6 by 4 and a triangle with base 4 and height 3 are attached. What is the total area?

28 m²

30 m²

34 m²

36 m²

Area = rectangle + triangle = (6 times 4) + ½ times (4 times 3) = 24 + 6 = 30 m². Option A gives 28, option C gives 34, option D gives 36, none of which correctly apply both area formulas.

Explanation

Area = rectangle + triangle = (6 times 4) + ½ times (4 times 3) = 24 + 6 = 30 m². Option A gives 28, option C gives 34, option D gives 36, none of which correctly apply both area formulas.

13) A right triangle with base 9 m and height 12 m is attached to a rectangle 9 m by 4 m. What is the total area?

80 m²

90 m²

120 m²

132 m²

Area = triangle + rectangle = ½ times (9 times 12) + (9 times 4) = 54 + 36 = 90 m². Option A gives 80, option C gives 120, option D gives 132, none of which correctly compute both component areas.

Explanation

Area = triangle + rectangle = ½ times (9 times 12) + (9 times 4) = 54 + 36 = 90 m². Option A gives 80, option C gives 120, option D gives 132, none of which correctly compute both component areas.

14) A rectangle 14 by 5 has a triangle with base 14 and height 3 attached on top. What is the total area?

90 m²

91 m²

96 m²

98 m²

Area = rectangle + triangle = (14 times 5) + ½ times (14 times 3) = 70 + 21 = 91 m². Option A gives 90, option C gives 96, and option D gives 98, none of which correctly compute both component areas.

Explanation

Area = rectangle + triangle = (14 times 5) + ½ times (14 times 3) = 70 + 21 = 91 m². Option A gives 90, option C gives 96, and option D gives 98, none of which correctly compute both component areas.

15) A square 8 by 8 has a right triangle with base 8 and height 6 attached on one side. What is the total area?

76 cm²

80 cm²

88 cm²

96 cm²

Area = square + triangle = (8 times 8) + ½ times (8 times 6) = 64 + 24 = 88 cm². Option A gives 76, option B gives 80, and option D gives 96, none of which correctly combine the square and triangle areas.

Explanation

Area = square + triangle = (8 times 8) + ½ times (8 times 6) = 64 + 24 = 88 cm². Option A gives 76, option B gives 80, and option D gives 96, none of which correctly combine the square and triangle areas.

16) A rectangular window 15 by 10 cm has a circular pane of radius 5 cm cut out. What is the remaining area?

70 cm²

71.5 cm²

75 cm²

80 cm²

Area = rectangle minus circle = (15 times 10) minus π times 5² = 150 minus 3.14159 times 25 = 150 minus 78.54 = 71.46 ≈ 71.5 cm². Option A gives 70, option C gives 75, option D gives 80, none of which correctly apply the circle area formula with radius 5.

Explanation

Area = rectangle minus circle = (15 times 10) minus π times 5² = 150 minus 3.14159 times 25 = 150 minus 78.54 = 71.46 ≈ 71.5 cm². Option A gives 70, option C gives 75, option D gives 80, none of which correctly apply the circle area formula with radius 5.

17) A rectangle 10 m by 6 m has a semicircle with diameter 10 m on top. What is the approximate total area?

80.4 m²

85.7 m²

90.2 m²

99.3 m²

Area = rectangle + semicircle = (10 times 6) + ½ times π times 5² = 60 + ½ times 3.14159 times 25 = 60 + 39.27 = 99.27 ≈ 99.3 m². Option A gives 80.4, option B gives 85.7, and option C gives 90.2, none of which correctly add the rectangle and semicircle areas.

Explanation

Area = rectangle + semicircle = (10 times 6) + ½ times π times 5² = 60 + ½ times 3.14159 times 25 = 60 + 39.27 = 99.27 ≈ 99.3 m². Option A gives 80.4, option B gives 85.7, and option C gives 90.2, none of which correctly add the rectangle and semicircle areas.

18) A square 14 by 14 has a quarter circle of radius 7 removed. What is the remaining area?

157.5 cm²

145.5 cm²

160.8 cm²

130.1 cm²

Area = square minus quarter circle = 14² minus ¼ times π times 7² = 196 minus ¼ times 3.14159 times 49 = 196 minus 38.48 = 157.52 ≈ 157.5 cm². Option B gives 145.5, option C gives 160.8, option D gives 130.1, none of which correctly apply the quarter circle formula with radius 7.

Explanation

Area = square minus quarter circle = 14² minus ¼ times π times 7² = 196 minus ¼ times 3.14159 times 49 = 196 minus 38.48 = 157.52 ≈ 157.5 cm². Option B gives 145.5, option C gives 160.8, option D gives 130.1, none of which correctly apply the quarter circle formula with radius 7.

19) The total area of a composite figure equals the sum of its non-overlapping parts.

True

False

The answer is True. A composite figure is made up of two or more simple shapes joined together. To find the total area, calculate each individual component area and add them together, ensuring no region is counted twice. When a shape is removed from another, subtract its area instead. The principle always holds as long as the parts do not overlap.

Explanation

The answer is True. A composite figure is made up of two or more simple shapes joined together. To find the total area, calculate each individual component area and add them together, ensuring no region is counted twice. When a shape is removed from another, subtract its area instead. The principle always holds as long as the parts do not overlap.

20) A trapezoid with base 12 m, top 8 m, height 5 m has a semicircle with diameter 8 m on top. What is the approximate total area?

75.1 m²

100.6 m²

120.6 m²

140 m²

Area = trapezoid + semicircle = ½ times (12+8) times 5 + ½ times π times 4² = ½ times 20 times 5 + ½ times 3.14159 times 16 = 50 + 25.13 = 75.13 ≈ 75.1 m². Option B gives 100.6, option C gives 120.6, option D gives 140, all of which overestimate the total.

Explanation

Area = trapezoid + semicircle = ½ times (12+8) times 5 + ½ times π times 4² = ½ times 20 times 5 + ½ times 3.14159 times 16 = 50 + 25.13 = 75.13 ≈ 75.1 m². Option B gives 100.6, option C gives 120.6, option D gives 140, all of which overestimate the total.