10-Volume of Pyramids and Cones - Kuta Software - Free Printable

Let's solve each of these pyramid and cone volume problems step by step. The volume of a pyramid or cone is given by the formula:

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

We’ll apply this formula to each figure, identifying the base area and height.

---

1) Triangular Pyramid (Tetrahedron)

- Base: Right triangle with legs 7 m and 7 m.
- Height of pyramid: 7 m (from apex to base)

Step 1: Find base area

The base is a right triangle:
$$
\text{Base Area} = \frac{1}{2} \times 7 \times 7 = \frac{49}{2} = 24.5 \text{ m}^2
$$

Step 2: Use volume formula
$$
V = \frac{1}{3} \times 24.5 \times 7 = \frac{171.5}{3} \approx 57.2 \text{ m}^3
$$

✔ Answer: 57.2 m³

---

2) Square Pyramid

- Base: Square with side 5 m
- Height of pyramid: 4 m (vertical from apex to center of base)
- Slant height is 3 m (not needed for volume)

Step 1: Base area
$$
\text{Base Area} = 5 \times 5 = 25 \text{ m}^2
$$

Step 2: Volume
$$
V = \frac{1}{3} \times 25 \times 4 = \frac{100}{3} \approx 33.3 \text{ m}^3
$$

✔ Answer: 33.3 m³

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3) Triangular Pyramid (with equilateral triangle base?)

- Base: Triangle with two sides 11 cm, and height from vertex to base = 12 cm?
Wait — actually, the diagram shows a triangle with base 11 cm and height 12 cm? Let’s interpret carefully.

Looking at the image:
- Base triangle has base = 11 cm, and height = 12 cm (perpendicular from apex to base).
- The pyramid has height = 12 cm (from top to base plane).

Wait — actually, the height of the pyramid is labeled as 12 cm, and the base triangle has a base of 11 cm and a height of 11 cm?

But looking closely: the base triangle has two sides marked 11 cm, and a perpendicular from one vertex to the base is shown as 11 cm? That might be confusing.

Actually, let's re-analyze:

It looks like:
- The base is an isosceles triangle with base 11 cm and height 11 cm (inside the base triangle).
- The pyramid height (from apex to base plane) is 12 cm.

So:

Step 1: Base area
$$
\text{Base Area} = \frac{1}{2} \times 11 \times 11 = \frac{121}{2} = 60.5 \text{ cm}^2
$$

Step 2: Volume
$$
V = \frac{1}{3} \times 60.5 \times 12 = \frac{726}{3} = 242 \text{ cm}^3
$$

✔ Answer: 242.0 cm³

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4) Triangular Pyramid (Triangular Base)

- Base: Triangle with sides 2 in, 5 in, and a height of 3 in (from vertex to base)
- Height of pyramid: 5 in (labeled as slant edge? Wait — no, the vertical height from apex to base is 5 in?)

Wait — the diagram shows a triangular base with base 5 in and height 3 in? And the pyramid height is 5 in?

Yes — it appears:
- Base triangle: base = 5 in, height = 3 in → area = $ \frac{1}{2} \times 5 \times 3 = 7.5 $ in²
- Height of pyramid = 5 in

Step 1: Base area
$$
\text{Base Area} = \frac{1}{2} \times 5 \times 3 = 7.5 \text{ in}^2
$$

Step 2: Volume
$$
V = \frac{1}{3} \times 7.5 \times 5 = \frac{37.5}{3} = 12.5 \text{ in}^3
$$

✔ Answer: 12.5 in³

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5) Rectangular Pyramid

- Base: Rectangle with dimensions 8.3 yd and 12 yd
- Height of pyramid: 11 yd (from apex to base)

Step 1: Base area
$$
\text{Base Area} = 8.3 \times 12 = 99.6 \text{ yd}^2
$$

Step 2: Volume
$$
V = \frac{1}{3} \times 99.6 \times 11 = \frac{1095.6}{3} = 365.2 \text{ yd}^3
$$

✔ Answer: 365.2 yd³

---

6) Square Pyramid

- Base: Square with side 5.2 m
- Height of pyramid: 3 m (vertical from apex to center of base)

Step 1: Base area
$$
\text{Base Area} = 5.2 \times 5.2 = 27.04 \text{ m}^2
$$

Step 2: Volume
$$
V = \frac{1}{3} \times 27.04 \times 3 = 27.04 \text{ m}^3
$$

✔ Answer: 27.0 m³

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✔ Final Answers:

1) 57.2 m³
2) 33.3 m³
3) 242.0 cm³
4) 12.5 in³
5) 365.2 yd³
6) 27.0 m³

All answers are rounded to the nearest tenth as requested.

Let me know if you'd like a visual explanation or help with any specific figure!