Y9. Shape & Space. Volumes of Prisms & Cylinders - Maths with David - Free Printable

Let's solve each part of Question 1: *Calculate the volume of each prism below.*

The volume of a prism is given by:

$$
\text{Volume} = \text{Base Area} \times \text{Height (or length of the prism)}
$$

We'll go through each figure one by one.

---

(a) Rectangular Prism

Dimensions: 3 cm × 4 cm × 6 cm

This is a rectangular prism, so:
$$
V = l \times w \times h = 3 \times 4 \times 6 = 72 \text{ cm}^3
$$

✔ Answer: 72 cm³

---

(b) Triangular Prism

Base triangle: right triangle with legs 5 cm and 6 cm
Length of prism: 20 cm

First, find area of triangular base:
$$
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 5 = 15 \text{ cm}^2
$$

Now volume:
$$
V = \text{Base Area} \times \text{Length} = 15 \times 20 = 300 \text{ cm}^3
$$

✔ Answer: 300 cm³

---

(c) Triangular Prism

Triangle base: base = 7 m, height = 5 m
Length of prism = 4 m

Area of triangle:
$$
A = \frac{1}{2} \times 7 \times 5 = 17.5 \text{ m}^2
$$

Volume:
$$
V = 17.5 \times 4 = 70 \text{ m}^3
$$

✔ Answer: 70 m³

---

(d) Rectangular Prism (Cuboid)

Dimensions: 1.1 m × 40 cm × 2 m

⚠️ Units must be consistent! Convert all to meters.

- 40 cm = 0.4 m

So:
$$
V = 1.1 \times 0.4 \times 2 = 0.88 \text{ m}^3
$$

✔ Answer: 0.88 m³

---

(e) Triangular Prism

Right triangle base: base = 7 cm, height = 8 cm
Length of prism = 5.5 cm

Area of triangle:
$$
A = \frac{1}{2} \times 7 \times 8 = 28 \text{ cm}^2
$$

Volume:
$$
V = 28 \times 5.5 = 154 \text{ cm}^3
$$

✔ Answer: 154 cm³

---

(f) Parallelogram-based Prism (Trapezoidal or Rhombic Base?)

Wait — this looks like a parallelogram-shaped base, but actually it’s a prism with a parallelogram cross-section.

But looking at the dimensions:
- Base side = 9 cm
- Height of parallelogram = 6 cm
- Length of prism = 12 cm

So, area of the parallelogram base:
$$
A = \text{base} \times \text{height} = 9 \times 6 = 54 \text{ cm}^2
$$

Volume:
$$
V = 54 \times 12 = 648 \text{ cm}^3
$$

✔ Answer: 648 cm³

---

(g) Triangular Prism

Right triangle base: sides 5 cm and 12 cm (right angle between them), hypotenuse 13 cm
Length of prism = 22 cm

Area of triangle:
$$
A = \frac{1}{2} \times 5 \times 12 = 30 \text{ cm}^2
$$

Volume:
$$
V = 30 \times 22 = 660 \text{ cm}^3
$$

✔ Answer: 660 cm³

---

(h) Trapezoidal Prism? Wait — looks like a trapezium-based prism

But let's examine:
The base is a trapezoid? Actually, from the diagram, it appears to be a trapezoidal prism with a trapezoid as the base.

But wait — the base has:
- Parallel sides: 25 cm and 10 cm?
- Height of trapezoid = 8 cm
- Length of prism = 40 cm

Wait — no. Let’s interpret carefully.

Actually, the cross-section is a trapezoid with:
- Two parallel sides: 25 cm and 10 cm
- Height of trapezoid = 8 cm
- The length of the prism (depth) = 40 cm

So, area of trapezoid:
$$
A = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}
= \frac{1}{2} \times (25 + 10) \times 8 = \frac{1}{2} \times 35 \times 8 = 140 \text{ cm}^2
$$

Volume:
$$
V = 140 \times 40 = 5600 \text{ cm}^3
$$

✔ Answer: 5600 cm³

---

(i) Triangular Prism

Triangle base: base = 14 cm, height = 9 cm
Length of prism = 15 cm

Area of triangle:
$$
A = \frac{1}{2} \times 14 \times 9 = 63 \text{ cm}^2
$$

Volume:
$$
V = 63 \times 15 = 945 \text{ cm}^3
$$

✔ Answer: 945 cm³

---

## ✔ Final Answers Summary:

| Part | Volume |
|------|--------|
| (a) | 72 cm³ |
| (b) | 300 cm³ |
| (c) | 70 m³ |
| (d) | 0.88 m³ |
| (e) | 154 cm³ |
| (f) | 648 cm³ |
| (g) | 660 cm³ |
| (h) | 5600 cm³ |
| (i) | 945 cm³ |

---

🔍 Key Notes:

- For rectangular prisms: $ V = lwh $
- For triangular prisms: $ V = \left(\frac{1}{2} \times b \times h_{\text{triangle}}\right) \times L $
- For parallelogram prisms: $ V = (\text{base} \times \text{height}) \times L $
- For trapezoidal prisms: $ V = \left(\frac{1}{2}(a+b)h\right) \times L $

Always check units and ensure consistency!

Let me know if you'd like diagrams or step-by-step visual explanations!