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. Volume = length × width × height
$$
V = 3 \times 4 \times 6 = 72 \text{ cm}^3
$$

✔ Answer: 72 cm³

---

(b) Triangular Prism

Right triangle base: legs = 5 cm and 6 cm, length = 20 cm

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

Now multiply by the length (20 cm):
$$
V = 15 \times 20 = 300 \text{ cm}^3
$$

✔ Answer: 300 cm³

---

(c) Triangular Prism

Triangle base: base = 7 m, height = 5 m, length = 4 m

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

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

✔ Answer: 70 m³

---

(d) Rectangular Prism (Cube-like but not cube)

Dimensions: 1.1 m × 40 cm × 2 m

⚠️ Units differ! Convert all to same unit — let’s use meters.

- 40 cm = 0.4 m

So dimensions: 1.1 m × 0.4 m × 2 m

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

✔ Answer: 0.88 m³

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(e) Triangular Prism

Triangle base: base = 7 cm, height = 8 cm, length = 5.5 cm

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

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

✔ Answer: 154 cm³

---

(f) Parallelogram-based Prism (Prism with parallelogram base)

Base: 9 cm × 6 cm (height), length = 12 cm

Wait — this is a prism with a parallelogram base, so we need the area of the base.

But note: the base is a parallelogram with base 9 cm and height 6 cm (perpendicular height).

So:
$$
\text{Base Area} = \text{base} \times \text{height} = 9 \times 6 = 54 \text{ cm}^2
$$

Length (depth) of prism = 12 cm

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

✔ Answer: 648 cm³

---

(g) Triangular Prism

Right triangle base: legs = 12 cm and 5 cm, hypotenuse = 13 cm, length = 22 cm

Check: 5² + 12² = 25 + 144 = 169 = 13² → right triangle ✔

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

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

✔ Answer: 660 cm³

---

(h) Trapezoidal Prism? Wait — looks like a trapezoidal prism or parallelogram-based?

Actually, the cross-section is a trapezoid or perhaps a parallelogram?

Wait — it shows two parallel sides: 25 cm and 40 cm? No — wait.

Looking at the diagram:
It appears that the cross-section is a trapezoid with bases 25 cm and 40 cm? But no — the side shows 8 cm height, and the top and bottom are 25 cm and 40 cm? Wait, actually, the figure seems to be a prism with a trapezoidal base.

Wait — looking closely: The base has a horizontal length of 25 cm, and the other side is 40 cm? That doesn't make sense.

Wait — actually, the side view shows a trapezoid with parallel sides 25 cm and 40 cm, and height 8 cm, and the length of the prism is 10 cm?

But the labels:
- Top edge: 25 cm
- Bottom edge: 40 cm
- Height between them: 8 cm
- Depth (length of prism): 10 cm

So yes — trapezoidal prism.

Area of trapezoid:
$$
A = \frac{1}{2} \times (b_1 + b_2) \times h = \frac{1}{2} \times (25 + 40) \times 8 = \frac{1}{2} \times 65 \times 8 = 260 \text{ cm}^2
$$

Then volume:
$$
V = 260 \times 10 = 2600 \text{ cm}^3
$$

✔ Answer: 2600 cm³

---

(i) Triangular Prism

Triangle base: base = 14 cm, height = 9 cm, length = 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

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✔ Final Answers:

| 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) | 2600 cm³ |
| (i) | 945 cm³ |

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🔍 Summary of Methods Used:

- For rectangular prisms: $ V = l \times w \times h $
- For triangular prisms: $ V = \left(\frac{1}{2} \times b \times h_{\text{triangle}}\right) \times L $
- For parallelogram or trapezoidal prisms: $ V = (\text{Base Area}) \times \text{Length} $

Make sure units are consistent!

Let me know if you'd like diagrams or explanations for any specific shape!