Let's solve each of the volume problems step by step. We'll use the appropriate volume formulas for each shape.
---
1) Cone
- Radius (r) = 14 ft
- Height (h) = 18 ft
- Volume of a cone:
$$
V = \frac{1}{3} \pi r^2 h
$$
$$
V = \frac{1}{3} \pi (14)^2 (18) = \frac{1}{3} \pi (196)(18) = \frac{1}{3} \pi (3528) = 1176\pi \text{ ft}^3
$$
Volume = $ 1176\pi $ ft³
---
2) Sphere
- Diameter = 10 yd → Radius (r) = 5 yd
- Volume of a sphere:
$$
V = \frac{4}{3} \pi r^3
$$
$$
V = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3}\pi \approx 166.67\pi \text{ yd}^3
$$
Volume = $ \frac{500}{3}\pi $ yd³
---
3) Rectangular Prism
- Length = 10 yd, Width = 5 yd, Height = 5 yd
- Volume:
$$
V = l \times w \times h = 10 \times 5 \times 5 = 250 \text{ yd}^3
$$
Volume = 250 yd³
---
4) Triangular Prism
- Base triangle: base = 12 yd, height = 16 yd
- Length of prism = 20 yd
- Volume of a prism = Area of base × length
Area of triangle = $ \frac{1}{2} \times 12 \times 16 = 96 $ yd²
$$
V = 96 \times 20 = 1920 \text{ yd}^3
$$
Volume = 1920 yd³
---
5) Rectangular Prism
- Dimensions: 6 ft × 6 ft × 4 ft
- Volume:
$$
V = 6 \times 6 \times 4 = 144 \text{ ft}^3
$$
Volume = 144 ft³
---
6) Cylinder
- Radius (r) = 8 in, Height (h) = 9 in
- Volume:
$$
V = \pi r^2 h = \pi (8)^2 (9) = \pi (64)(9) = 576\pi \text{ in}^3
$$
Volume = $ 576\pi $ in³
---
7) Hemisphere
- Radius (r) = 10 ft
- Volume of a hemisphere = $ \frac{2}{3} \pi r^3 $
$$
V = \frac{2}{3} \pi (10)^3 = \frac{2}{3} \pi (1000) = \frac{2000}{3}\pi \approx 666.67\pi \text{ ft}^3
$$
Volume = $ \frac{2000}{3}\pi $ ft³
---
8) Rectangular Prism
- Dimensions: 5 yd × 10 yd × 4 yd
- Volume:
$$
V = 5 \times 10 \times 4 = 200 \text{ yd}^3
$$
Volume = 200 yd³
---
9) Cone
- Radius (r) = 8 ft, Height (h) = 18 ft
- Volume:
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (8)^2 (18) = \frac{1}{3} \pi (64)(18) = \frac{1}{3} \pi (1152) = 384\pi \text{ ft}^3
$$
Volume = $ 384\pi $ ft³
---
✔ Final Answers:
| Problem | Volume |
|--------|--------|
| 1) | $ 1176\pi $ ft³ |
| 2) | $ \frac{500}{3}\pi $ yd³ |
| 3) | 250 yd³ |
| 4) | 1920 yd³ |
| 5) | 144 ft³ |
| 6) | $ 576\pi $ in³ |
| 7) | $ \frac{2000}{3}\pi $ ft³ |
| 8) | 200 yd³ |
| 9) | $ 384\pi $ ft³ |
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Note: Unless specified otherwise, we leave answers in terms of $ \pi $ for exact values.
Let me know if you'd like decimal approximations!
Parent Tip: Review the logic above to help your child master the concept of volume of shapes worksheet.
