Here are the step-by-step solutions for each problem on the worksheet.
1. Rectangle
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Formula: Area = base $\times$ height
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Calculation: $10.9 \text{ m} \times 7.5 \text{ m}$
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Result: $81.75 \text{ m}^2$
2. Regular Pentagon
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Formula: Area = $\frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$
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Step 1 (Perimeter): A pentagon has 5 sides. $5 \times 13.2 \text{ yd} = 66 \text{ yd}$.
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Step 2 (Area): $\frac{1}{2} \times 66 \times 3.5$
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Calculation: $33 \times 3.5 = 115.5$
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Result: $115.5 \text{ yd}^2$
3. Regular Hexagon
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Formula: Area = $\frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$
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Step 1 (Perimeter): A hexagon has 6 sides. $6 \times 12 \text{ cm} = 72 \text{ cm}$.
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Step 2 (Area): $\frac{1}{2} \times 72 \times 5\sqrt{3}$
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Calculation: $36 \times 5\sqrt{3} = 180\sqrt{3}$
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Decimal Approximation: $180 \times 1.732 \approx 311.77$
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Result: $180\sqrt{3} \text{ cm}^2$ (or approx. $311.8 \text{ cm}^2$)
4. Triangle
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Formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$
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Calculation: $\frac{1}{2} \times 22 \text{ in} \times 16 \text{ in}$
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Simplify: $11 \times 16$
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Result: $176 \text{ in}^2$
5. Composite Figure (Rectangle + Semicircle)
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Part 1: Rectangle
* Area = $32 \text{ ft} \times 18 \text{ ft} = 576 \text{ ft}^2$
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Part 2: Semicircle
* Diameter = $18 \text{ ft}$, so Radius ($r$) = $9 \text{ ft}$.
* Area of full circle = $\pi \times 9^2 = 81\pi$.
* Area of semicircle = $\frac{81\pi}{2} = 40.5\pi \approx 127.23 \text{ ft}^2$.
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Total Area: $576 + 127.23$
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Result: $703.23 \text{ ft}^2$
6. Composite Figure (Triangle + Semicircle)
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Part 1: Triangle
* Base = $9 \text{ mm}$.
* Height = Total height ($12$) minus radius. Radius is half of $9$ ($4.5$). So, Height = $12 - 4.5 = 7.5 \text{ mm}$.
* Area = $\frac{1}{2} \times 9 \times 7.5 = 33.75 \text{ mm}^2$.
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Part 2: Semicircle
* Radius = $4.5 \text{ mm}$.
* Area = $\frac{\pi \times 4.5^2}{2} = \frac{20.25\pi}{2} \approx 31.81 \text{ mm}^2$.
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Total Area: $33.75 + 31.81$
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Result: $65.56 \text{ mm}^2$
7. Composite Figure (Rectangle + Triangle)
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Part 1: Rectangle
* Area = $10 \text{ km} \times 6 \text{ km} = 60 \text{ km}^2$.
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Part 2: Triangle
* Base = $6 \text{ km}$ (matches the rectangle side).
* Height = $4.8 \text{ km}$.
* Area = $\frac{1}{2} \times 6 \times 4.8 = 14.4 \text{ km}^2$.
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Total Area: $60 + 14.4$
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Result: $74.4 \text{ km}^2$
8. Composite Figure (Square + Triangle)
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Part 1: Square
* Area = $16 \text{ cm} \times 16 \text{ cm} = 256 \text{ cm}^2$.
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Part 2: Triangle
* Base = $16 \text{ cm}$ (matches the square side).
* Height = $8.5 \text{ cm}$.
* Area = $\frac{1}{2} \times 16 \times 8.5 = 8 \times 8.5 = 68 \text{ cm}^2$.
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Total Area: $256 + 68$
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Result: $324 \text{ cm}^2$
Final Answer:
1. 81.75 m²
2. 115.5 yd²
3. 180√3 cm² (approx. 311.8 cm²)
4. 176 in²
5. 703.23 ft²
6. 65.56 mm²
7. 74.4 km²
8. 324 cm²
Parent Tip: Review the logic above to help your child master the concept of area of composite figures worksheet 7th grade answers.
