Volume of Pyramids - Free Printable

Let's solve the problem step by step.

We are given three square-based pyramids and asked to calculate their volumes. The formula for the volume of a square-based pyramid is:

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

Where:
- Base Area = side² (since it's a square base)
- Height = perpendicular height from the apex to the base

---

🔷 Pyramid 1 (Green)

- Base side = 5 cm
- Height = 6 cm

$$
\text{Base Area} = 5 \times 5 = 25 \, \text{cm}^2
$$
$$
V = \frac{1}{3} \times 25 \times 6 = \frac{150}{3} = 50 \, \text{cm}^3
$$

✔ Volume = 50 cm³

---

🔷 Pyramid 2 (Pink)

- Base side = 6 cm
- Height = 8 cm

$$
\text{Base Area} = 6 \times 6 = 36 \, \text{cm}^2
$$
$$
V = \frac{1}{3} \times 36 \times 8 = \frac{288}{3} = 96 \, \text{cm}^3
$$

✔ Volume = 96 cm³

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🔷 Pyramid 3 (Blue) – This one looks like a frustum, but let’s analyze carefully.

Wait — this shape has two different bases:
- Bottom base = 12 cm × 12 cm
- Top base = 10 cm × 10 cm
- Height = 8 cm

This is not a full pyramid, but a frustum of a square pyramid — the top part has been cut off.

So we need to use the volume of a frustum formula:

$$
V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2})
$$

Where:
- $ h $ = height of the frustum = 8 cm
- $ A_1 $ = area of the lower base = $ 12 \times 12 = 144 \, \text{cm}^2 $
- $ A_2 $ = area of the upper base = $ 10 \times 10 = 100 \, \text{cm}^2 $

Now plug in:

$$
V = \frac{1}{3} \times 8 \times (144 + 100 + \sqrt{144 \times 100})
$$
$$
= \frac{8}{3} \times (244 + \sqrt{14400})
$$
$$
= \frac{8}{3} \times (244 + 120)
$$
$$
= \frac{8}{3} \times 364 = \frac{2912}{3} \approx 970.67 \, \text{cm}^3
$$

✔ Volume ≈ 970.67 cm³

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✔ Final Answers:

1. Green Pyramid: 50 cm³
2. Pink Pyramid: 96 cm³
3. Blue Frustum: ≈ 970.67 cm³

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📌 Summary:

| Shape | Volume |
|------|--------|
| Green Square Pyramid | 50 cm³ |
| Pink Square Pyramid | 96 cm³ |
| Blue Frustum | ~970.67 cm³ |

Let me know if you'd like a diagram explanation or how to derive the frustum formula!