Let's solve the problem step by step for each of the three solids:
sphere,
cone, and
square-based pyramid. We'll find the volume of each in
cm³, providing:
- (a) Exact form
- (b) Rounded to 2 decimal places
---
🔹 1. Sphere
Given: Radius $ r = 7 $ cm
Formula for volume of a sphere:
$$
V = \frac{4}{3} \pi r^3
$$
#### (a) Exact form:
$$
V = \frac{4}{3} \pi (7)^3 = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi \text{ cm}^3
$$
#### (b) Approximate value:
$$
V \approx \frac{1372}{3} \times 3.1416 \approx 457.333 \times 3.1416 \approx 1436.76 \text{ cm}^3
$$
✔ Sphere Volume:
- (a) $ \frac{1372}{3} \pi $ cm³
- (b)
1436.76 cm³
---
🔹 2. Cone
Given:
- Radius $ r = 4 $ cm
- Height $ h = 13 $ cm
Formula for volume of a cone:
$$
V = \frac{1}{3} \pi r^2 h
$$
#### (a) Exact form:
$$
V = \frac{1}{3} \pi (4)^2 (13) = \frac{1}{3} \pi (16)(13) = \frac{1}{3} \pi (208) = \frac{208}{3} \pi \text{ cm}^3
$$
#### (b) Approximate value:
$$
V \approx \frac{208}{3} \times 3.1416 \approx 69.333 \times 3.1416 \approx 218.18 \text{ cm}^3
$$
✔ Cone Volume:
- (a) $ \frac{208}{3} \pi $ cm³
- (b)
218.18 cm³
---
🔹 3. Square-based Pyramid
Given:
- Base side length = 60 cm
- Height = 30 cm
Formula for volume of a pyramid:
$$
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
$$
Base is a square:
$$
\text{Base Area} = 60 \times 60 = 3600 \text{ cm}^2
$$
So,
$$
V = \frac{1}{3} \times 3600 \times 30 = 1200 \times 30 = 36,000 \text{ cm}^3
$$
This is already an exact number.
✔ Pyramid Volume:
- (a) $ 36,000 $ cm³
- (b)
36,000.00 cm³ (no rounding needed)
---
✔ Final Answers:
| Solid | Exact Form | Rounded (2 dp) |
|------|------------|----------------|
|
Sphere | $ \frac{1372}{3} \pi $ cm³ |
1436.76 cm³ |
|
Cone | $ \frac{208}{3} \pi $ cm³ |
218.18 cm³ |
|
Square-based Pyramid | $ 36,000 $ cm³ |
36,000.00 cm³ |
---
Let me know if you'd like these visualized or explained further! 🦆🦆🦆
Parent Tip: Review the logic above to help your child master the concept of volumes of sphere cone and pyramid worksheet.
