Volume of Pyramids, Cones, and Spheres worksheet with 16 problems for calculating volumes of geometric shapes.

Let's solve each problem step by step using the formulas for volume of pyramids, cones, and spheres.

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Formulas:

1. Volume of a Pyramid:
$$
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
$$

2. Volume of a Cone:
$$
V = \frac{1}{3} \pi r^2 h
$$

3. Volume of a Sphere:
$$
V = \frac{4}{3} \pi r^3
$$

4. Volume of a Hemisphere (half sphere):
$$
V = \frac{2}{3} \pi r^3
$$

5. For a rectangular pyramid, base area = length × width
For triangular pyramid, base area = $ \frac{1}{2} \times \text{base} \times \text{height} $

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We'll go through each question:

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1) Square Pyramid

- Base: 4 cm × 4 cm → Area = $ 4 \times 4 = 16 \text{ cm}^2 $
- Height = 9 cm
- Volume = $ \frac{1}{3} \times 16 \times 9 = 48 \text{ cm}^3 $

✔ Answer: 48.00 cm³

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2) Triangular Pyramid (Tetrahedron)

- Base is triangle with base = 12 cm, height = 6 cm → Area = $ \frac{1}{2} \times 12 \times 6 = 36 \text{ cm}^2 $
- Height of pyramid = 15 cm
- Volume = $ \frac{1}{3} \times 36 \times 15 = 180 \text{ cm}^3 $

✔ Answer: 180.00 cm³

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3) Rectangular Pyramid

- Base: 5 cm × 7 cm → Area = $ 5 \times 7 = 35 \text{ cm}^2 $
- Height = 20 cm
- Volume = $ \frac{1}{3} \times 35 \times 20 = \frac{700}{3} \approx 233.33 \text{ cm}^3 $

✔ Answer: 233.33 cm³

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4) Rectangular Pyramid (with cube-like base)

- Base: 3 cm × 2 cm → Area = $ 3 \times 2 = 6 \text{ cm}^2 $
- Height = 7 cm
- Volume = $ \frac{1}{3} \times 6 \times 7 = 14 \text{ cm}^3 $

✔ Answer: 14.00 cm³

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5) Cone

- Radius = 3 cm, Height = 9 cm
- Volume = $ \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (9) = \frac{1}{3} \pi \times 9 \times 9 = 27\pi \approx 84.82 \text{ cm}^3 $

✔ Answer: 84.82 cm³

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6) Cone

- Radius = 1.5 cm, Height = 7 cm
- Volume = $ \frac{1}{3} \pi (1.5)^2 (7) = \frac{1}{3} \pi \times 2.25 \times 7 = \frac{1}{3} \pi \times 15.75 \approx 16.49 \text{ cm}^3 $

✔ Answer: 16.49 cm³

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7) Cone

- Radius = 10 cm (since diameter = 20 cm), Height = 24 cm
- Volume = $ \frac{1}{3} \pi (10)^2 (24) = \frac{1}{3} \pi \times 100 \times 24 = 800\pi \approx 2513.27 \text{ cm}^3 $

✔ Answer: 2513.27 cm³

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8) Cone

- Diameter = 7.2 mm → Radius = 3.6 mm
- Height = 7.7 mm
- Volume = $ \frac{1}{3} \pi (3.6)^2 (7.7) = \frac{1}{3} \pi \times 12.96 \times 7.7 \approx \frac{1}{3} \pi \times 100.032 \approx 104.72 \text{ mm}^3 $

✔ Answer: 104.72 mm³

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9) Sphere

- Radius = 10 cm
- Volume = $ \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi \times 1000 \approx 4188.79 \text{ cm}^3 $

✔ Answer: 4188.79 cm³

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10) Sphere

- Radius = 2 cm
- Volume = $ \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi \times 8 \approx 33.51 \text{ cm}^3 $

✔ Answer: 33.51 cm³

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11) Sphere

- Radius = 11 cm (diameter = 22 cm)
- Volume = $ \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi \times 1331 \approx 5575.28 \text{ cm}^3 $

✔ Answer: 5575.28 cm³

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12) Sphere

- Radius = 17 mm
- Volume = $ \frac{4}{3} \pi (17)^3 = \frac{4}{3} \pi \times 4913 \approx 20579.52 \text{ mm}^3 $

✔ Answer: 20579.52 mm³

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13) Hemisphere

- Radius = 7 cm
- Volume = $ \frac{2}{3} \pi (7)^3 = \frac{2}{3} \pi \times 343 \approx 718.38 \text{ cm}^3 $

✔ Answer: 718.38 cm³

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14) Hemisphere

- Radius = 4.2 cm (diameter = 8.4 cm)
- Volume = $ \frac{2}{3} \pi (4.2)^3 = \frac{2}{3} \pi \times 74.088 \approx 155.43 \text{ cm}^3 $

✔ Answer: 155.43 cm³

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15) Triangular Pyramid

- Base is triangle with base = 6 cm, height = 6 cm → Area = $ \frac{1}{2} \times 6 \times 6 = 18 \text{ cm}^2 $
- Height of pyramid = 15 cm
- Volume = $ \frac{1}{3} \times 18 \times 15 = 90 \text{ cm}^3 $

✔ Answer: 90.00 cm³

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16) Cone

- Radius = 4 mm, Height = 12 mm
- Volume = $ \frac{1}{3} \pi (4)^2 (12) = \frac{1}{3} \pi \times 16 \times 12 = 64\pi \approx 201.06 \text{ mm}^3 $

✔ Answer: 201.06 mm³

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✔ Final Answers:

| Problem | Volume (rounded to 2 decimal places) |
|--------|--------------------------------------|
| 1) | 48.00 cm³ |
| 2) | 180.00 cm³ |
| 3) | 233.33 cm³ |
| 4) | 14.00 cm³ |
| 5) | 84.82 cm³ |
| 6) | 16.49 cm³ |
| 7) | 2513.27 cm³ |
| 8) | 104.72 mm³ |
| 9) | 4188.79 cm³ |
| 10) | 33.51 cm³ |
| 11) | 5575.28 cm³ |
| 12) | 20579.52 mm³ |
| 13) | 718.38 cm³ |
| 14) | 155.43 cm³ |
| 15) | 90.00 cm³ |
| 16) | 201.06 mm³ |

> 🔍 Note: All calculations use $ \pi \approx 3.1416 $. Round as needed.

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