Geometry - Volume and Surface Area - Volume of Pyramids Cones and ... - Free Printable

Let's solve each problem step by step using the appropriate volume formulas.

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Volume Formulas:

- Pyramid:
$$
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
$$

- Cone:
$$
V = \frac{1}{3} \pi r^2 h
$$

- Sphere:
$$
V = \frac{4}{3} \pi r^3
$$

- Hemisphere:
$$
V = \frac{2}{3} \pi r^3
$$

We’ll use $ \pi \approx 3.14 $, and round answers to 2 decimal places where needed.

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1) Square Pyramid

- Base: 4 cm × 4 cm → Area = $ 4 \times 4 = 16 \text{ cm}^2 $
- Height = 9 cm
$$
V = \frac{1}{3} \times 16 \times 9 = \frac{144}{3} = 48.00 \text{ cm}^3
$$

✔ Volume = 48.00 cm³

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2) Triangular Pyramid (Tetrahedron)

- Base: Triangle with base 12 cm, height 7 cm → Area = $ \frac{1}{2} \times 12 \times 7 = 42 \text{ cm}^2 $
- Height of pyramid = 15 cm
$$
V = \frac{1}{3} \times 42 \times 15 = 210.00 \text{ cm}^3
$$

✔ Volume = 210.00 cm³

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3) Triangular Pyramid

- Base: Triangle with base 7 cm, height 5 cm → Area = $ \frac{1}{2} \times 7 \times 5 = 17.5 \text{ cm}^2 $
- Height of pyramid = 20 cm
$$
V = \frac{1}{3} \times 17.5 \times 20 = \frac{350}{3} \approx 116.67 \text{ cm}^3
$$

✔ Volume = 116.67 cm³

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4) Rectangular Pyramid

- Base: 3 cm × 2 cm → Area = $ 3 \times 2 = 6 \text{ cm}^2 $
- Height = 7 cm
$$
V = \frac{1}{3} \times 6 \times 7 = 14.00 \text{ cm}^3
$$

✔ Volume = 14.00 cm³

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5) Cone

- Radius = 3 cm, Height = 9 cm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 3^2 \times 9 = \frac{1}{3} \times 3.14 \times 9 \times 9 = 3.14 \times 27 = 84.78 \text{ cm}^3
$$

✔ Volume = 84.78 cm³

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6) Cone

- Radius = 1.5 cm, Height = 7 cm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times (1.5)^2 \times 7 = \frac{1}{3} \times 3.14 \times 2.25 \times 7
= 3.14 \times 0.75 \times 7 = 3.14 \times 5.25 = 16.49 \text{ cm}^3
$$

✔ Volume = 16.49 cm³

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7) Cone

- Radius = 10 cm (since diameter = 20 cm), Height = 24 cm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 10^2 \times 24 = \frac{1}{3} \times 3.14 \times 100 \times 24 = 3.14 \times 800 = 2512.00 \text{ cm}^3
$$

✔ Volume = 2512.00 cm³

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8) Cone

- Diameter = 7.2 mm → radius = 3.6 mm
- Height = 7.7 mm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times (3.6)^2 \times 7.7 = \frac{1}{3} \times 3.14 \times 12.96 \times 7.7
= 3.14 \times 4.32 \times 7.7 = 3.14 \times 33.264 = 104.43 \text{ mm}^3
$$

✔ Volume = 104.43 mm³

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9) Sphere

- Radius = 10 cm
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 10^3 = \frac{4}{3} \times 3.14 \times 1000 = 4186.67 \text{ cm}^3
$$

✔ Volume = 4186.67 cm³

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10) Sphere

- Radius = 2 cm
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 8 = \frac{4}{3} \times 25.12 = 33.49 \text{ cm}^3
$$

✔ Volume = 33.49 cm³

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11) Sphere

- Radius = 11 cm (diameter = 22 cm)
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 1331 = \frac{4}{3} \times 3.14 \times 1331 = 4.1867 \times 1331 \approx 5575.28 \text{ cm}^3
$$

Wait — let's compute:
$$
\frac{4}{3} \times 3.14 \times 1331 = \left( \frac{4 \times 3.14}{3} \right) \times 1331 = (4.1867) \times 1331 \approx 5575.28
$$

✔ Volume = 5575.28 cm³

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12) Sphere

- Radius = 17 mm
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times (17)^3 = \frac{4}{3} \times 3.14 \times 4913 = 4.1867 \times 4913 \approx 20578.46 \text{ mm}^3
$$

✔ Volume = 20578.46 mm³

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13) Hemisphere

- Radius = 7 cm
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times 343 = 2.0933 \times 343 \approx 718.38 \text{ cm}^3
$$

✔ Volume = 718.38 cm³

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14) Hemisphere

- Radius = 4.2 cm (diameter = 8.4 cm)
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times (4.2)^3 = \frac{2}{3} \times 3.14 \times 74.088 = 2.0933 \times 74.088 \approx 155.14 \text{ cm}^3
$$

✔ Volume = 155.14 cm³

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15) Triangular Pyramid

- Base: 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
$$
V = \frac{1}{3} \times 18 \times 15 = 90.00 \text{ cm}^3
$$

✔ Volume = 90.00 cm³

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16) Cone

- Radius = 4 mm, Height = 12 mm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 16 \times 12 = 3.14 \times \frac{192}{3} = 3.14 \times 64 = 200.96 \text{ mm}^3
$$

✔ Volume = 200.96 mm³

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✔ Final Answers:

| # | Volume |
|---|--------|
| 1 | 48.00 cm³ |
| 2 | 210.00 cm³ |
| 3 | 116.67 cm³ |
| 4 | 14.00 cm³ |
| 5 | 84.78 cm³ |
| 6 | 16.49 cm³ |
| 7 | 2512.00 cm³ |
| 8 | 104.43 mm³ |
| 9 | 4186.67 cm³ |
|10 | 33.49 cm³ |
|11 | 5575.28 cm³ |
|12 | 20578.46 mm³ |
|13 | 718.38 cm³ |
|14 | 155.14 cm³ |
|15 | 90.00 cm³ |
|16 | 200.96 mm³ |

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