Surface Area: Cylinders Textbook Exercise - Corbettmaths - Free Printable

Let's solve Question 1 and Question 2 step by step.

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🔷 Surface Area of a Cylinder Formula

The total surface area (SA) of a cylinder is given by:

$$
\text{SA} = 2\pi r^2 + 2\pi r h
$$

Where:
- $ r $ = radius of the base
- $ h $ = height of the cylinder
- $ \pi \approx 3.1416 $

This formula includes:
- $ 2\pi r^2 $: area of the two circular bases
- $ 2\pi r h $: lateral (side) surface area

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## ✔ Question 1: Work out the surface area of each cylinder. Give answers to 2 decimal places.

We’ll calculate each one using $ \pi \approx 3.1416 $.

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(a)

- Radius $ r = 3 $ cm
- Height $ h = 10 $ cm

$$
\text{SA} = 2\pi r^2 + 2\pi r h = 2\pi(3)^2 + 2\pi(3)(10)
= 2\pi(9) + 2\pi(30)
= 18\pi + 60\pi = 78\pi
$$

Now compute numerically:
$$
78\pi \approx 78 \times 3.1416 = 245.0448 \approx \boxed{245.04} \text{ cm}^2
$$

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(b)

- Radius $ r = 8 $ cm
- Height $ h = 5 $ cm

$$
\text{SA} = 2\pi(8)^2 + 2\pi(8)(5)
= 2\pi(64) + 2\pi(40)
= 128\pi + 80\pi = 208\pi
$$

Numerical value:
$$
208\pi \approx 208 \times 3.1416 = 653.4528 \approx \boxed{653.45} \text{ cm}^2
$$

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(c)

- Radius $ r = 14 $ cm
- Height $ h = 16 $ cm

$$
\text{SA} = 2\pi(14)^2 + 2\pi(14)(16)
= 2\pi(196) + 2\pi(224)
= 392\pi + 448\pi = 840\pi
$$

Numerical:
$$
840\pi \approx 840 \times 3.1416 = 2638.944 \approx \boxed{2638.94} \text{ cm}^2
$$

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(d)

- Radius $ r = 0.9 $ cm
- Height $ h = 1.7 $ cm

$$
\text{SA} = 2\pi(0.9)^2 + 2\pi(0.9)(1.7)
= 2\pi(0.81) + 2\pi(1.53)
= 1.62\pi + 3.06\pi = 4.68\pi
$$

Numerical:
$$
4.68\pi \approx 4.68 \times 3.1416 = 14.623 \approx \boxed{14.62} \text{ cm}^2
$$

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(e)

- Radius $ r = 3 $ m
- Height $ h = 2 $ m

$$
\text{SA} = 2\pi(3)^2 + 2\pi(3)(2)
= 2\pi(9) + 2\pi(6)
= 18\pi + 12\pi = 30\pi
$$

Numerical:
$$
30\pi \approx 30 \times 3.1416 = 94.248 \approx \boxed{94.25} \text{ m}^2
$$

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(f)

- Radius $ r = 4 $ cm
- Height $ h = 36 $ mm → convert to cm: $ 36 \text{ mm} = 3.6 \text{ cm} $

$$
\text{SA} = 2\pi(4)^2 + 2\pi(4)(3.6)
= 2\pi(16) + 2\pi(14.4)
= 32\pi + 28.8\pi = 60.8\pi
$$

Numerical:
$$
60.8\pi \approx 60.8 \times 3.1416 = 191.543 \approx \boxed{191.54} \text{ cm}^2
$$

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## ✔ Question 2: Surface area in terms of $ \pi $

We'll use the same formula, but leave answers in terms of $ \pi $.

But wait — the image only shows cylinders (a)–(f), and Question 2 says “the following cylinders” — but no new diagrams are shown. So likely, Question 2 refers to the same cylinders as Question 1, just asking for answers in terms of $ \pi $ instead.

So let’s do that:

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(a)

$$
\text{SA} = 2\pi(3)^2 + 2\pi(3)(10) = 18\pi + 60\pi = \boxed{78\pi} \text{ cm}^2
$$

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(b)

$$
\text{SA} = 2\pi(8)^2 + 2\pi(8)(5) = 128\pi + 80\pi = \boxed{208\pi} \text{ cm}^2
$$

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(c)

$$
\text{SA} = 2\pi(14)^2 + 2\pi(14)(16) = 392\pi + 448\pi = \boxed{840\pi} \text{ cm}^2
$$

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(d)

$$
\text{SA} = 2\pi(0.9)^2 + 2\pi(0.9)(1.7) = 1.62\pi + 3.06\pi = \boxed{4.68\pi} \text{ cm}^2
$$

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(e)

$$
\text{SA} = 2\pi(3)^2 + 2\pi(3)(2) = 18\pi + 12\pi = \boxed{30\pi} \text{ m}^2
$$

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(f)

$$
\text{SA} = 2\pi(4)^2 + 2\pi(4)(3.6) = 32\pi + 28.8\pi = \boxed{60.8\pi} \text{ cm}^2
$$

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## ✔ Final Answers

Question 1: (to 2 decimal places)

| Part | Surface Area |
|------|--------------|
| (a) | $ \boxed{245.04} \text{ cm}^2 $ |
| (b) | $ \boxed{653.45} \text{ cm}^2 $ |
| (c) | $ \boxed{2638.94} \text{ cm}^2 $ |
| (d) | $ \boxed{14.62} \text{ cm}^2 $ |
| (e) | $ \boxed{94.25} \text{ m}^2 $ |
| (f) | $ \boxed{191.54} \text{ cm}^2 $ |

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Question 2: In terms of $ \pi $

| Part | Surface Area |
|------|--------------|
| (a) | $ \boxed{78\pi} \text{ cm}^2 $ |
| (b) | $ \boxed{208\pi} \text{ cm}^2 $ |
| (c) | $ \boxed{840\pi} \text{ cm}^2 $ |
| (d) | $ \boxed{4.68\pi} \text{ cm}^2 $ |
| (e) | $ \boxed{30\pi} \text{ m}^2 $ |
| (f) | $ \boxed{60.8\pi} \text{ cm}^2 $ |

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✔ Summary:
- Always ensure units are consistent (e.g., convert mm to cm).
- Use $ \text{SA} = 2\pi r^2 + 2\pi r h $
- For Question 1: plug in $ \pi \approx 3.1416 $ and round to 2 decimal places.
- For Question 2: keep answer in terms of $ \pi $.

Let me know if you'd like a diagram or explanation of how the formula works!