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 = 8 yd
- Length of prism = 20 yd
- Volume of triangular prism:
$$
V = \text{Base Area} \times \text{Length}
$$
Base area = $ \frac{1}{2} \times 12 \times 8 = 48 $ yd²
$$
V = 48 \times 20 = 960 \text{ yd}^3
$$
Volume = 960 yd³
---
5) Cube / Rectangular Prism
- Dimensions: 6 ft × 6 ft × 8 ft
- Volume:
$$
V = 6 \times 6 \times 8 = 288 \text{ ft}^3
$$
Volume = 288 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 full sphere: $ \frac{4}{3} \pi r^3 $, so hemisphere is half of that:
$$
V = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
$$
$$
V = \frac{2}{3} \pi (10)^3 = \frac{2}{3} \pi (1000) = \frac{2000}{3}\pi \approx 666.67\pi \text{ ft}^3
$$
Volume = $ \frac{2000}{3}\pi $ ft³
---
8) Rectangular Prism
- Length = 10 yd, Width = 5 yd, Height = 5 yd
- Volume:
$$
V = 10 \times 5 \times 5 = 250 \text{ yd}^3
$$
Volume = 250 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) | 960 yd³ |
| 5) | 288 ft³ |
| 6) | $ 576\pi $ in³ |
| 7) | $ \frac{2000}{3}\pi $ ft³ |
| 8) | 250 yd³ |
| 9) | $ 384\pi $ ft³ |
> Note: For exact volumes, we leave answers in terms of $ \pi $ unless otherwise specified.
Let me know if you'd like decimal approximations!
Parent Tip: Review the logic above to help your child master the concept of volume worksheet 6th grade.
