To find the volume of pyramids and cones, we use specific formulas. Here is the step-by-step solution for each problem on the worksheet.
General Formulas:
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Volume of a Pyramid: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$
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Volume of a Cone: $V = \frac{1}{3} \times \pi \times r^2 \times h$ (where $r$ is radius and $h$ is height)
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1) Square Pyramid
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Base: Square with side 3 ft.
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Height: 11 ft.
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Base Area: $3 \times 3 = 9 \text{ sq ft}$.
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Volume: $\frac{1}{3} \times 9 \times 11 = 3 \times 11 = 33$.
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Answer: 33 cu ft
2) Rectangular Pyramid
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Base: Rectangle ($2 \text{ cm} \times 8 \text{ cm}$).
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Height: 10 cm.
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Base Area: $2 \times 8 = 16 \text{ sq cm}$.
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Volume: $\frac{1}{3} \times 16 \times 10 = \frac{160}{3} \approx 53.33$.
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Answer: 53.33 cu cm
3) Rectangular Pyramid
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Base: Rectangle ($12 \text{ cm} \times 6 \text{ cm}$).
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Height: 10 cm.
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Base Area: $12 \times 6 = 72 \text{ sq cm}$.
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Volume: $\frac{1}{3} \times 72 \times 10 = 24 \times 10 = 240$.
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Answer: 240 cu cm
4) Rectangular Pyramid
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Base: Rectangle ($4 \text{ mm} \times 10 \text{ mm}$).
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Height: 12 mm.
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Base Area: $4 \times 10 = 40 \text{ sq mm}$.
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Volume: $\frac{1}{3} \times 40 \times 12 = 40 \times 4 = 160$.
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Answer: 160 cu mm
5) Rectangular Pyramid
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Base: Rectangle ($4 \text{ in} \times 10 \text{ in}$).
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Height: 12 in.
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Base Area: $4 \times 10 = 40 \text{ sq in}$.
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Volume: $\frac{1}{3} \times 40 \times 12 = 40 \times 4 = 160$.
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Answer: 160 cu in
6) Triangular Pyramid
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Base: Triangle with base 13 in and height 5 in.
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Pyramid Height: 10 in.
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Base Area: $\frac{1}{2} \times 13 \times 5 = 32.5 \text{ sq in}$.
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Volume: $\frac{1}{3} \times 32.5 \times 10 = \frac{325}{3} \approx 108.33$.
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Answer: 108.33 cu in
7) Square Pyramid
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Base: Square with side 5 mm.
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Height: 13 mm.
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Base Area: $5 \times 5 = 25 \text{ sq mm}$.
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Volume: $\frac{1}{3} \times 25 \times 13 = \frac{325}{3} \approx 108.33$.
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Answer: 108.33 cu mm
8) Cone
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Radius: The diagram shows the full diameter is 4 yd, so the radius ($r$) is 2 yd.
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Height: 11 yd.
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Volume: $\frac{1}{3} \times \pi \times 2^2 \times 11$.
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Calculation: $\frac{1}{3} \times \pi \times 4 \times 11 = \frac{44\pi}{3} \approx 46.08$.
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Answer: 46.08 cu yd
9) Cone
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Radius: 2 yd.
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Height: 10 yd.
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Volume: $\frac{1}{3} \times \pi \times 2^2 \times 10$.
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Calculation: $\frac{1}{3} \times \pi \times 4 \times 10 = \frac{40\pi}{3} \approx 41.89$.
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Answer: 41.89 cu yd
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Final Answer:
1) 33 cu ft
2) 53.33 cu cm
3) 240 cu cm
4) 160 cu mm
5) 160 cu in
6) 108.33 cu in
7) 108.33 cu mm
8) 46.08 cu yd
9) 41.89 cu yd
Parent Tip: Review the logic above to help your child master the concept of surface areas of pyramids and cones worksheet.
