Volume & Surface Area - Free Printable

Let's solve each part of the problem step by step.

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1. Shapes made of 1 cm cubes

We are given three 3D shapes built from 1 cm × 1 cm × 1 cm cubes. We need to find:

- Volume: Number of small cubes (each has volume = 1 cm³)
- Surface Area: Total area of all exposed faces (each face is 1 cm²)

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#### a)
This shape looks like a rectangular block:
- Dimensions: 3 cm (length) × 2 cm (width) × 3 cm (height)
- Count the cubes:
$ 3 \times 2 \times 3 = 18 $ cubes

✔ Volume = 18 cm³

Now, surface area:
Use formula for cuboid:
$$
\text{Surface Area} = 2(lw + lh + wh)
$$
Where:
- $ l = 3 $, $ w = 2 $, $ h = 3 $

$$
= 2(3×2 + 3×3 + 2×3) = 2(6 + 9 + 6) = 2(21) = 42 \text{ cm}^2
$$

✔ Surface Area = 42 cm²

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#### b)
This is a cube made of smaller cubes:
- Each side has 4 cubes → so it’s a 4×4×4 cube
- Total cubes: $ 4 × 4 × 4 = 64 $

✔ Volume = 64 cm³

Surface area:
$$
\text{Surface Area} = 6 × (\text{side})^2 = 6 × 4^2 = 6 × 16 = 96 \text{ cm}^2
$$

✔ Surface Area = 96 cm²

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#### c)
This is a long rectangular prism:
- Dimensions: 1 cm high, 1 cm wide, 10 cm long → 1×1×10
- Cubes: $ 1 × 1 × 10 = 10 $

✔ Volume = 10 cm³

Surface area:
$$
= 2(lw + lh + wh) = 2(10×1 + 10×1 + 1×1) = 2(10 + 10 + 1) = 2(21) = 42 \text{ cm}^2
$$

✔ Surface Area = 42 cm²

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✔ Summary for Part 1:

| Shape | Volume (cm³) | Surface Area (cm²) |
|-------|---------------|---------------------|
| a) | 18 | 42 |
| b) | 64 | 96 |
| c) | 10 | 42 |

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2. Work out volume and surface area of each cuboid

We now use formulas:
- Volume = $ l × w × h $
- Surface Area = $ 2(lw + lh + wh) $

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#### a)
Dimensions: 7 cm × 4 cm × 5 cm

✔ Volume = $ 7 × 4 × 5 = 140 $ cm³

✔ Surface Area = $ 2(7×4 + 7×5 + 4×5) = 2(28 + 35 + 20) = 2(83) = 166 $ cm²

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#### b)
Dimensions: 6 mm × 3 mm × 2 mm

✔ Volume = $ 6 × 3 × 2 = 36 $ mm³

✔ Surface Area = $ 2(6×3 + 6×2 + 3×2) = 2(18 + 12 + 6) = 2(36) = 72 $ mm²

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#### c)
Dimensions: 50 cm × 2 m × 4 m

⚠️ Units are mixed! Convert to same unit.

Convert meters to centimeters:
- 2 m = 200 cm
- 4 m = 400 cm

So dimensions: 50 cm × 200 cm × 400 cm

✔ Volume = $ 50 × 200 × 400 = 4,000,000 $ cm³
Or in m³:
$ 0.5 \text{ m} × 2 \text{ m} × 4 \text{ m} = 4 $ m³
But since question uses cm, keep as 4,000,000 cm³

✔ Surface Area = $ 2(lw + lh + wh) $

$$
= 2(50×200 + 50×400 + 200×400)
= 2(10,000 + 20,000 + 80,000)
= 2(110,000) = 220,000 \text{ cm}^2
$$

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✔ Summary for Part 2:

| Shape | Volume | Surface Area |
|-------|----------------|------------------|
| a) | 140 cm³ | 166 cm² |
| b) | 36 mm³ | 72 mm² |
| c) | 4,000,000 cm³ | 220,000 cm² |

> Note: For part 2c, you could also express in meters:
> - Volume = 4 m³
> - Surface Area = 220,000 cm² = 22 m² (since 1 m² = 10,000 cm² → 220,000 ÷ 10,000 = 22 m²)

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✔ Final Answers:

#### Part 1:
- a) Volume = 18 cm³, Surface Area = 42 cm²
- b) Volume = 64 cm³, Surface Area = 96 cm²
- c) Volume = 10 cm³, Surface Area = 42 cm²

#### Part 2:
- a) Volume = 140 cm³, Surface Area = 166 cm²
- b) Volume = 36 mm³, Surface Area = 72 mm²
- c) Volume = 4,000,000 cm³ (or 4 m³), Surface Area = 220,000 cm² (or 22 m²)

Let me know if you'd like these shown in diagrams or with visual explanations!