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

Let's solve each part of Question 1, where we are asked to calculate the volume of each prism.

The general formula for the volume of a prism is:

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

We will go through each shape one by one.

---

(a) Rectangular Prism

Dimensions:
- Length = 6 cm
- Width = 4 cm
- Height = 3 cm

This is a rectangular prism. The base area is length × width, and height is the depth.

$$
\text{Volume} = 6 \times 4 \times 3 = 72\ \text{cm}^3
$$

✔ Answer: 72 cm³

---

(b) Triangular Prism

Right triangle base:
- Base = 6 cm
- Height = 5 cm
- Length (depth) = 20 cm

Area of triangular base:
$$
\frac{1}{2} \times 6 \times 5 = 15\ \text{cm}^2
$$

Volume:
$$
15 \times 20 = 300\ \text{cm}^3
$$

✔ Answer: 300 cm³

---

(c) Triangular Prism

Triangle base:
- Base = 7 m
- Height = 5 m
- Length = 4 m

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

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

✔ Answer: 70 m³

---

(d) Rectangular Prism (Cuboid)

Dimensions:
- Length = 40 cm
- Width = 1.1 m = 110 cm (convert to same unit!)
- Height = 2 m = 200 cm

Wait! Units are mixed. Let's convert everything to centimeters.

- 1.1 m = 110 cm
- 2 m = 200 cm

So volume:
$$
40 \times 110 \times 200 = 880,000\ \text{cm}^3
$$

Alternatively, convert to meters:
- 40 cm = 0.4 m
- 1.1 m
- 2 m

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

But since input was in cm/m mix, let’s keep it consistent.

✔ Answer: 880,000 cm³ or 0.88 m³
*(Usually, we prefer same units — so 0.88 m³ might be better)*

But let's check:
Since two dimensions are in meters and one in cm, best to convert all to cm.

Yes:
40 cm × 110 cm × 200 cm = 880,000 cm³

✔ Answer: 880,000 cm³

---

(e) Triangular Prism

Triangle base:
- Base = 7 cm
- Height = 8 cm
- Length = 5.5 cm

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

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

✔ Answer: 154 cm³

---

(f) Parallelogram-based Prism

This is a parallelogram prism (like a slanted rectangular prism).
We can treat it as a prism with a parallelogram base.

But the base is a parallelogram with:
- Base = 9 cm
- Height = 6 cm (perpendicular height)

So area of base:
$$
\text{Base} \times \text{Height} = 9 \times 6 = 54\ \text{cm}^2
$$

Length of prism = 12 cm

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

✔ Answer: 648 cm³

---

(g) Triangular Prism

Right triangle base:
- Legs: 5 cm and 12 cm
- Hypotenuse = 13 cm (given, confirms right triangle)
- Length = 22 cm

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

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

✔ Answer: 660 cm³

---

(h) Trapezoidal Prism

Cross-section is a trapezium (trapezoid).

Trapezoid base:
- Parallel sides: 25 cm and 40 cm? Wait — looking at diagram:

Actually, from the figure:
- Two parallel sides: 25 cm and 40 cm? No — wait, the side marked 25 cm and 40 cm appear to be the lengths along the base, but the height of the trapezoid is 8 cm (vertical dashed line), and the length of the prism is 10 cm.

Wait — the trapezoid has:
- Two parallel sides: 25 cm and 40 cm
- Height (distance between them) = 8 cm
- The prism extends 10 cm (depth)

So base area = area of trapezoid:
$$
\text{Area} = \frac{1}{2} \times (a + b) \times h = \frac{1}{2} \times (25 + 40) \times 8 = \frac{1}{2} \times 65 \times 8 = 260\ \text{cm}^2
$$

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

✔ Answer: 2600 cm³

---

(i) Triangular Prism

Triangle base:
- Base = 14 cm
- Height = 9 cm
- Length = 15 cm

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

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

✔ Answer: 945 cm³

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✔ Final Answers Summary:

| Part | Volume |
|------|--------|
| (a) | 72 cm³ |
| (b) | 300 cm³ |
| (c) | 70 m³ |
| (d) | 880,000 cm³ |
| (e) | 154 cm³ |
| (f) | 648 cm³ |
| (g) | 660 cm³ |
| (h) | 2600 cm³ |
| (i) | 945 cm³ |

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🔍 Notes:

- Always ensure units are consistent.
- For prisms: Volume = Base Area × Length (or height of prism).
- Triangular bases: use $ \frac{1}{2} \times b \times h $
- Trapezoidal base: $ \frac{1}{2}(a+b) \times h $
- Parallelogram: $ b \times h $

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