Volume of Pyramids, Cones, and Spheres worksheet with 16 problems for calculating the volume of geometric shapes.
Worksheet titled "Volume of Pyramids, Cones, and Spheres" with 16 problems showing various 3D shapes including pyramids, cones, and spheres, each with dimensions and a space to calculate volume.
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Show Answer Key & Explanations
Step-by-step solution for: Geometry - Volume and Surface Area - Volume of Pyramids Cones and ...
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Show Answer Key & Explanations
Step-by-step solution for: Geometry - Volume and Surface Area - Volume of Pyramids Cones and ...
Let's solve each problem step by step using the appropriate 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.
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- 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
---
- Radius = 17 mm
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times (17)^3 = \frac{4}{3} \times 3.14 \times 4913
= 4.1867 \times 4913 \approx 20574.57 \text{ mm}^3
$$
✔ Volume = 20574.57 mm³
---
- Radius = 7 cm
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times 343 = 2.0933 \times 343 \approx 717.67 \text{ cm}^3
$$
✔ Volume = 717.67 cm³
---
- Radius = 4.2 cm (since diameter = 8.4 cm)
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times (4.2)^3 = \frac{2}{3} \times 3.14 \times 74.088
= 2.0933 \times 74.088 \approx 155.10 \text{ cm}^3
$$
✔ Volume = 155.10 cm³
---
- 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
---
- Radius = 4 mm, Height = 12 mm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 16 \times 12 = 3.14 \times 64 = 200.96 \text{ mm}^3
$$
✔ Volume = 200.96 mm³
---
| Problem | 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) | 20574.57 mm³ |
| 13) | 717.67 cm³ |
| 14) | 155.10 cm³ |
| 15) | 90.00 cm³ |
| 16) | 200.96 mm³ |
---
Let me know if you'd like these in a printable format or need help understanding any specific shape!
---
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.
---
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
---
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
---
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
---
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
---
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
---
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
---
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
---
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
---
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
---
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
---
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
---
12) Sphere
- Radius = 17 mm
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times (17)^3 = \frac{4}{3} \times 3.14 \times 4913
= 4.1867 \times 4913 \approx 20574.57 \text{ mm}^3
$$
✔ Volume = 20574.57 mm³
---
13) Hemisphere
- Radius = 7 cm
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times 343 = 2.0933 \times 343 \approx 717.67 \text{ cm}^3
$$
✔ Volume = 717.67 cm³
---
14) Hemisphere
- Radius = 4.2 cm (since diameter = 8.4 cm)
$$
V = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times (4.2)^3 = \frac{2}{3} \times 3.14 \times 74.088
= 2.0933 \times 74.088 \approx 155.10 \text{ cm}^3
$$
✔ Volume = 155.10 cm³
---
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
---
16) Cone
- Radius = 4 mm, Height = 12 mm
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 16 \times 12 = 3.14 \times 64 = 200.96 \text{ mm}^3
$$
✔ Volume = 200.96 mm³
---
✔ Final Answers:
| Problem | 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) | 20574.57 mm³ |
| 13) | 717.67 cm³ |
| 14) | 155.10 cm³ |
| 15) | 90.00 cm³ |
| 16) | 200.96 mm³ |
---
Let me know if you'd like these in a printable format or need help understanding any specific shape!
Parent Tip: Review the logic above to help your child master the concept of volumes of sphere cone and pyramid worksheet.
