Here is the step-by-step solution for each of the 9 compound shapes. We’ll break each shape into simpler geometric parts (triangles, rectangles, circles, semicircles), compute their areas, and then add or subtract as needed.
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1) Triangle with a circle cut out
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Triangle: base = 29 ft, height = 29 ft
Area = (1/2) × base × height = (1/2) × 29 × 29 =
420.5 ft²
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Circle: radius = 9 ft
Area = πr² = π × 81 ≈ 3.1416 × 81 ≈
254.47 ft²
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Shaded area = Triangle – Circle = 420.5 – 254.47 ≈
166.0 ft²
✔ Answer: 166.0 ft²
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2) Right triangle + semicircle
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Triangle: legs = 11 in and 5 in
Area = (1/2) × 11 × 5 =
27.5 in²
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Semicircle: radius = 5 in
Area = (1/2) × π × r² = (1/2) × π × 25 ≈ (1/2) × 78.54 ≈
39.27 in²
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Total area = 27.5 + 39.27 ≈
66.8 in²
✔ Answer: 66.8 in²
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3) Square with inscribed circle (shaded = square minus circle)
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Square: side = 16 in
Area = 16 × 16 =
256 in²
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Circle: radius = 8 in (since diameter = 16)
Area = π × 8² = π × 64 ≈ 201.06 in²
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Shaded area = 256 – 201.06 ≈
54.9 in²
✔ Answer: 54.9 in²
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4) “C” shape — rectangle minus smaller rectangle
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Outer rectangle: 14 m × 11 m =
154 m²
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Inner rectangle (cutout): 7 m × 7 m =
49 m²
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Shaded area = 154 – 49 =
105 m²
✔ Answer: 105.0 m²
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5) Rectangle with semicircular cutout
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Rectangle: 11 cm × ? — Wait, height isn’t labeled. But looking at the semicircle, it’s cut from the right side, and the radius is 4 cm → so the height must be 8 cm (diameter).
So rectangle: 11 cm × 8 cm =
88 cm²
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Semicircle: radius = 4 cm
Area = (1/2) × π × 16 ≈ (1/2) × 50.27 ≈
25.13 cm²
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Shaded area = 88 – 25.13 ≈
62.9 cm²
✔ Answer: 62.9 cm²
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6) Rectangle + triangle
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Rectangle: 12 m × 12 m =
144 m²
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Triangle: base = 6 m, height = 12 m (same as rectangle height)
Area = (1/2) × 6 × 12 =
36 m²
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Total area = 144 + 36 =
180 m²
✔ Answer: 180.0 m²
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7) Two triangles + semicircle
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Each triangle: base = 2 yd, height = 5 yd
Area of one = (1/2) × 2 × 5 =
5 yd²
Two triangles = 5 × 2 =
10 yd²
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Semicircle: radius = 2 yd
Area = (1/2) × π × 4 ≈ (1/2) × 12.566 ≈
6.283 yd²
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Total area = 10 + 6.283 ≈
16.3 yd²
✔ Answer: 16.3 yd²
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8) Large square minus small square
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Large square: 6 ft × 6 ft =
36 ft²
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Small square: 3 ft × 3 ft =
9 ft²
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Shaded area = 36 – 9 =
27 ft²
✔ Answer: 27.0 ft²
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9) Rectangle + triangle
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Rectangle: 20 yd × 15 yd =
300 yd²
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Triangle: base = 20 yd, height = 10 yd
Area = (1/2) × 20 × 10 =
100 yd²
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Total area = 300 + 100 =
400 yd²
✔ Answer: 400.0 yd²
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##
✔ Final Answers:
1)
166.0 ft²
2)
66.8 in²
3)
54.9 in²
4)
105.0 m²
5)
62.9 cm²
6)
180.0 m²
7)
16.3 yd²
8)
27.0 ft²
9)
400.0 yd²
All answers are rounded to
one decimal place where necessary (though many were whole numbers). Let me know if you’d like diagrams or further breakdowns!
Parent Tip: Review the logic above to help your child master the concept of area mixed shapes worksheet.
