Volume of rectangular pyramids worksheet with answer key.

To solve the problem of finding the volume of each rectangular pyramid, we use the formula for the volume of a pyramid:

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

For a rectangular pyramid, the base is a rectangle, so the base area is:

$$
\text{Base Area} = \text{length} \times \text{width}
$$

Then:
$$
V = \frac{1}{3} \times (\text{length} \times \text{width}) \times \text{height}
$$

Let’s go through each problem step by step and verify the given answers.

---

1)

- Length = 13 cm
- Width = 6 cm
- Height = 14 cm

$$
V = \frac{1}{3} \times (13 \times 6) \times 14 = \frac{1}{3} \times 78 \times 14 = \frac{1092}{3} = 364 \text{ cm}^3
$$

✔ Correct

---

2)

- Length = 21 m
- Width = 21 m
- Height = 29 m

$$
V = \frac{1}{3} \times (21 \times 21) \times 29 = \frac{1}{3} \times 441 \times 29 = \frac{12789}{3} = 4263 \text{ m}^3
$$

✔ Correct

---

3)

- Length = 32 ft
- Width = 17 ft
- Height = 24 ft

$$
V = \frac{1}{3} \times (32 \times 17) \times 24 = \frac{1}{3} \times 544 \times 24 = \frac{12960}{3} = 4320 \text{ ft}^3
$$

But the answer key says 4352 ft³ — this is incorrect.

Wait! Let's double-check:

$$
32 \times 17 = 544 \\
544 \times 24 = 12,960 \\
12,960 ÷ 3 = 4,320
$$

So correct volume is 4320 ft³, not 4352.

✘ Answer key has an error here.

---

4)

- Length = 18 in
- Width = 5 in
- Height = 11 in

$$
V = \frac{1}{3} \times (18 \times 5) \times 11 = \frac{1}{3} \times 90 \times 11 = \frac{990}{3} = 330 \text{ in}^3
$$

✔ Correct

---

5)

- Length = 22 m
- Width = 21 m
- Height = 35 m

$$
V = \frac{1}{3} \times (22 \times 21) \times 35 = \frac{1}{3} \times 462 \times 35 = \frac{16170}{3} = 5390 \text{ m}^3
$$

✔ Correct

---

6)

- Length = 8 cm
- Width = 3 cm
- Height = 10 cm

$$
V = \frac{1}{3} \times (8 \times 3) \times 10 = \frac{1}{3} \times 24 \times 10 = \frac{240}{3} = 80 \text{ cm}^3
$$

✔ Correct

---

7)

- Length = 32 ft
- Width = 32 ft
- Height = 50 ft

$$
V = \frac{1}{3} \times (32 \times 32) \times 50 = \frac{1}{3} \times 1024 \times 50 = \frac{51200}{3} \approx 17066.67 \text{ ft}^3
$$

✔ Correct

---

8)

- Length = 15 in
- Width = 15 in
- Height = 19 in

$$
V = \frac{1}{3} \times (15 \times 15) \times 19 = \frac{1}{3} \times 225 \times 19 = \frac{4275}{3} = 1425 \text{ in}^3
$$

✔ Correct

---

9)

- Length = 27 m
- Width = 23 m
- Height = 28 m

$$
V = \frac{1}{3} \times (27 \times 23) \times 28 = \frac{1}{3} \times 621 \times 28 = \frac{17388}{3} = 5796 \text{ m}^3
$$

✔ Correct

---

✔ Summary:

All answers are correct except for #3.

#### ✘ Problem 3: Error in Answer Key
Given dimensions:
- Base: 32 ft × 17 ft
- Height: 24 ft

Correct calculation:
$$
V = \frac{1}{3} \times 32 \times 17 \times 24 = \frac{1}{3} \times 12960 = 4320 \text{ ft}^3
$$

But the answer key says 4352 ft³ → Incorrect

---

✔ Final Answer:

The correct volumes are:

| Problem | Volume |
|--------|--------|
| 1) | 364 cm³ |
| 2) | 4263 m³ |
| 3) | 4320 ft³ *(not 4352)* |
| 4) | 330 in³ |
| 5) | 5390 m³ |
| 6) | 80 cm³ |
| 7) | 17066.67 ft³ |
| 8) | 1425 in³ |
| 9) | 5796 m³ |

> 🔴 Note: The answer key incorrectly lists 4352 ft³ for problem 3. The correct volume is 4320 ft³.