Volume of Cylinders

A cylindrical water tank has a radius of 1.5 m and a height of 2 m. Before you learn the formula, make a prediction: how much water do you think it can hold? What information would you need to work this out?

Think about how you calculated the volume of rectangular boxes. How might you extend that idea to a cylinder, which has a circular base instead of a rectangular one?

Every prism has its volume calculated the same way: Volume = base area × height. A cylinder is simply a prism with a circular base, so:

• Base area = area of a circle = $\pi r^2$
• Volume = base area × height = $\pi r^2 \times h$

This is why $V = \pi r^2 h$ — we're not memorising a random formula; we're applying the prism rule to a circular cross-section.

Step-by-step derivation:

1. Any prism: $V = A_{\text{base}} \times h$
2. For a cylinder, $A_{\text{base}} = \pi r^2$ (area of a circle)
3. Substituting: $V = \pi r^2 \times h$
4. Written compactly: $$V = \pi r^2 h$$

Key trap: Always use the radius ($r$), not the diameter ($d$). If given $d$, calculate $r = d \div 2$ first, then substitute.

When you are given the radius and height directly, substitute straight into $V = \pi r^2 h$:

1. Write the formula: $V = \pi r^2 h$
2. Identify $r$ and $h$ from the question
3. Substitute and calculate $r^2$ first
4. Multiply by $\pi$ then by $h$
5. Write the answer with correct units (cm³, m³, etc.)

Example: $r = 5$ cm, $h = 8$ cm.

$V = \pi \times 5^2 \times 8 = \pi \times 25 \times 8 = 200\pi \approx 628.3 \text{ cm}^3$

Tip: Leave your answer as a multiple of $\pi$ (e.g., $200\pi$) for an exact answer, or use the $\pi$ button on your calculator to get a decimal approximation.

When a question gives you the diameter instead of the radius, always halve it first before substituting.

Step 1: $r = d \div 2$

Step 2: $V = \pi r^2 h$

Example: $d = 10$ cm, $h = 6$ cm.

$r = 10 \div 2 = 5$ cm

$V = \pi \times 5^2 \times 6 = \pi \times 25 \times 6 = 150\pi \approx 471.2 \text{ cm}^3$

If you know the volume and one dimension, rearrange the formula to find the other.

Finding height given V and r:

Start with $V = \pi r^2 h$ and divide both sides by $\pi r^2$:

$$h = \frac{V}{\pi r^2}$$

Example: $V = 314.2$ cm³, $r = 5$ cm. $h = \frac{314.2}{\pi \times 25} = \frac{314.2}{78.54} \approx 4$ cm.

Finding radius given V and h:

Start with $V = \pi r^2 h$, divide by $\pi h$, then take the square root:

$$r = \sqrt{\frac{V}{\pi h}}$$

Example: $V = 502.7$ cm³, $h = 10$ cm. $r = \sqrt{\frac{502.7}{\pi \times 10}} = \sqrt{\frac{502.7}{31.42}} \approx \sqrt{16} = 4$ cm.

Set a timer for 5 minutes. Give answers to 1 decimal place where needed.

1. 1 r = 3 cm, h = 7 cm. Find the volume.

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   V = π × 9 × 7 = 63π ≈ 197.9 cm³

2. 2 r = 5 cm, h = 4 cm. Find the volume.

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   V = π × 25 × 4 = 100π ≈ 314.2 cm³

3. 3 d = 8 cm, h = 5 cm. Find the volume.

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   r = 4 cm. V = π × 16 × 5 = 80π ≈ 251.3 cm³

4. 4 d = 14 cm, h = 10 cm. Find the volume.

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   r = 7 cm. V = π × 49 × 10 = 490π ≈ 1539.4 cm³

5. 5 r = 2 cm, h = 15 cm. Find the volume.

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   V = π × 4 × 15 = 60π ≈ 188.5 cm³

6. 6 r = 6 cm, h = 6 cm. Find the volume.

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   V = π × 36 × 6 = 216π ≈ 678.6 cm³

7. 7 V = 628.3 cm³, r = 5 cm. Find the height.

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   h = 628.3 ÷ (π × 25) ≈ 628.3 ÷ 78.54 ≈ 8.0 cm

8. 8 V = 1130.97 cm³, h = 10 cm. Find the radius.

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   r = √(1130.97 ÷ (π × 10)) ≈ √(36) = 6 cm

9. 9 r = 1.5 m, h = 2 m. Find volume in m³ and capacity in L.

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   V = π × 2.25 × 2 = 4.5π ≈ 14.137 m³. Capacity: 14,137 L

10. 10 r = 4 cm, h = 12 cm. Volume in cm³ and capacity in mL.

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    V = π × 16 × 12 = 192π ≈ 603.2 cm³ = 603.2 mL