Here are the step-by-step solutions to find the surface area for each figure.
1. Rectangular Prism
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Formula: $2(lw + lh + wh)$
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Dimensions: Length = 8 ft, Width = 4 ft, Height = 4 ft
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Calculation:
* Top/Bottom: $8 \times 4 = 32$
* Front/Back: $8 \times 4 = 32$
* Sides: $4 \times 4 = 16$
* Total: $2 \times (32 + 32 + 16) = 2 \times 80 = 160$
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Answer: 160 sq ft
2. Triangular Prism
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Formula: Area of 2 triangular bases + Area of 3 rectangular sides
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Triangle Base: Base = 5 cm, Height = 4 cm. Area = $\frac{1}{2} \times 5 \times 4 = 10$. Two triangles = $20$.
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Rectangular Sides: The perimeter of the triangle is $5 + 5 + 6 = 16$ cm. The length of the prism is 4 cm.
* Side Area: $16 \times 4 = 64$.
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Total: $20 + 64 = 84$
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Answer: 84 sq cm
3. Cube
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Formula: $6 \times s^2$
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Side: 10 mm
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Calculation: $6 \times (10 \times 10) = 6 \times 100 = 600$
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Answer: 600 sq mm
4. Rectangular Prism
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Formula: $2(lw + lh + wh)$
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Dimensions: Length = 9 in, Width = 2 in, Height = 6 in
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Calculation:
* Top/Bottom: $9 \times 2 = 18$
* Front/Back: $9 \times 6 = 54$
* Sides: $2 \times 6 = 12$
* Total: $2 \times (18 + 54 + 12) = 2 \times 84 = 168$
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Answer: 168 sq in
5. Square Pyramid
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Formula: Base Area + Lateral Area ($\frac{1}{2} \times \text{Perimeter} \times \text{Slant Height}$)
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Base: Square with side 4 ft. Area = $4 \times 4 = 16$.
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Lateral Area: Perimeter = $4 \times 4 = 16$ ft. Slant Height = 6 ft.
* Area = $\frac{1}{2} \times 16 \times 6 = 48$.
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Total: $16 + 48 = 64$
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Answer: 64 sq ft
6. Triangular Pyramid (Regular Tetrahedron)
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Formula: Since all faces are identical equilateral triangles, calculate one face and multiply by 4. Or use Base + Lateral.
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Base Triangle: Base = 3 cm, Height = 8 cm. Area = $\frac{1}{2} \times 3 \times 8 = 12$.
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Lateral Area: There are 3 identical side triangles. $3 \times 12 = 36$.
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Total: $12 (\text{base}) + 36 (\text{sides}) = 48$.
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Answer: 48 sq cm
7. Triangular Pyramid
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Formula: Base Area + Sum of 3 Side Triangles
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Base Triangle: Base = 8.7 mm, Height = 10 mm.
* Area = $\frac{1}{2} \times 8.7 \times 10 = 43.5$.
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Side Triangles: There are 3 sides. Each has a base of 8.7 mm and a slant height of 12 mm.
* Area of one side = $\frac{1}{2} \times 8.7 \times 12 = 52.2$.
* Total for 3 sides = $3 \times 52.2 = 156.6$.
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Total: $43.5 + 156.6 = 200.1$
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Answer: 200.1 sq mm
8. Square Pyramid
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Formula: Base Area + Lateral Area
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Base: Square with side 9 in. Area = $9 \times 9 = 81$.
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Lateral Area: Perimeter = $4 \times 9 = 36$ in. Slant Height = 9 in.
* Area = $\frac{1}{2} \times 36 \times 9 = 18 \times 9 = 162$.
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Total: $81 + 162 = 243$
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Answer: 243 sq in
Final Answer:
1. 160 sq ft
2. 84 sq cm
3. 600 sq mm
4. 168 sq in
5. 64 sq ft
6. 48 sq cm
7. 200.1 sq mm
8. 243 sq in
Parent Tip: Review the logic above to help your child master the concept of prism and pyramid worksheet.
