Volume of 3D Shapes — Prisms, Cylinders, Pyramids, and Cones

Grade: 8–9 | Topic: Geometry

What You Will Learn

By the end of this page, you will be able to apply the correct volume formula for any common 3D shape — prism, cylinder, pyramid, cone, and sphere — and solve both straightforward and multi-step problems involving volume.

Theory

What Is Volume?

Volume measures the amount of three-dimensional space inside a shape. It is measured in cubic units: cm³, m³, mm³.

Prisms and Cylinders: $V = B \times h$

Both prisms and cylinders share the same logic: volume equals the area of the base times the height.

$V = B \times h$

where $B$ is the area of the base and $h$ is the height (perpendicular distance between the two bases).

Shape	Base area $B$	Volume formula
Rectangular prism	$l \times w$	$V = lwh$
Triangular prism	$\frac{1}{2}bh_{\text{tri}}$	$V = \frac{1}{2}b h_{\text{tri}} \times h$
Cylinder	$\pi r^2$	$V = \pi r^2 h$

Pyramids and Cones: $V = \dfrac{1}{3}Bh$

Pyramids and cones hold exactly $\dfrac{1}{3}$ of the volume of a prism or cylinder with the same base and height.

$V = \frac{1}{3} \times B \times h$

Shape	Base area $B$	Volume formula
Square pyramid	$s^2$	$V = \frac{1}{3}s^2 h$
Cone	$\pi r^2$	$V = \frac{1}{3}\pi r^2 h$

Sphere: $V = \dfrac{4}{3}\pi r^3$

$V = \frac{4}{3}\pi r^3$

where $r$ is the radius. Note: this formula uses the cube of the radius, not the square.

Worked Examples

Example 1: Volume of a Rectangular Prism

Problem: A fish tank is 60 cm long, 30 cm wide, and 40 cm tall. How many litres of water does it hold? (1 litre = 1000 cm³)

Step 1: Apply $V = lwh$. $V = 60 \times 30 \times 40 = 72{,}000 \text{ cm}^3$

Step 2: Convert to litres. $72{,}000 \div 1{,}000 = 72 \text{ litres}$

Answer: The tank holds 72 litres.

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Example 2: Volume of a Cylinder

Problem: Find the volume of a can with radius 4 cm and height 12 cm. Use $\pi \approx 3.14$.

Step 1: Apply $V = \pi r^2 h$. $V = 3.14 \times 4^2 \times 12 = 3.14 \times 16 \times 12$

Step 2: Calculate. $3.14 \times 192 = 602.88 \text{ cm}^3$

Answer: Volume $\approx \mathbf{602.9 \text{ cm}^3}$

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Example 3: Volume of a Cone

Problem: A traffic cone has a base radius of 15 cm and a height of 60 cm. Find its volume.

Step 1: Apply $V = \dfrac{1}{3}\pi r^2 h$. $V = \frac{1}{3} \times 3.14 \times 15^2 \times 60$

Step 2: Calculate step by step. $15^2 = 225$ $V = \frac{1}{3} \times 3.14 \times 225 \times 60 = \frac{1}{3} \times 42{,}390 = 14{,}130 \text{ cm}^3$

Answer: Volume $= \mathbf{14{,}130 \text{ cm}^3}$

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Example 4: Volume of a Sphere

Problem: A basketball has a radius of 12 cm. Find its volume. Leave your answer in terms of $\pi$.

Step 1: Apply $V = \dfrac{4}{3}\pi r^3$. $V = \frac{4}{3} \times \pi \times 12^3$

Step 2: Calculate $12^3 = 1728$. $V = \frac{4}{3} \times 1728\pi = \frac{6912\pi}{3} = 2304\pi \text{ cm}^3$

Answer: Volume $= \mathbf{2304\pi} \approx 7{,}238.2 \text{ cm}^3$

Common Mistakes

Mistake 1: Using Diameter Instead of Radius

❌ A cylinder with diameter 8 cm: $V = \pi \times 8^2 \times h$.

✅ Radius $= 8 \div 2 = 4$ cm. $V = \pi \times 4^2 \times h$. Always halve the diameter first.

Mistake 2: Forgetting the $\dfrac{1}{3}$ Factor for Pyramids and Cones

❌ $V_{\text{cone}} = \pi r^2 h$ (same as cylinder).

✅ $V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h$. A cone holds one-third as much as a cylinder of the same dimensions.

Mistake 3: Squaring the Radius in the Sphere Formula

❌ $V = \dfrac{4}{3}\pi r^2$ (sphere formula with $r^2$).

✅ The sphere formula uses $r^3$: $V = \dfrac{4}{3}\pi r^3$.

Practice Problems

Try these on your own before checking the answers:

1. Find the volume of a rectangular prism with length 8 m, width 3 m, height 5 m.
2. A cylinder has radius 6 cm and height 9 cm. Find its volume in terms of $\pi$.
3. A square pyramid has a base of side 10 cm and height 15 cm. Find its volume.
4. Find the volume of a sphere with radius 5 cm. Use $\pi \approx 3.14$.
5. A cylindrical water tower has diameter 6 m and height 20 m. How many cubic metres of water can it hold?

Click to see answers

1. $V = 8 \times 3 \times 5 = \mathbf{120 \text{ m}^3}$
2. $V = \pi \times 36 \times 9 = \mathbf{324\pi \text{ cm}^3} \approx 1017.9 \text{ cm}^3$
3. $V = \frac{1}{3} \times 10^2 \times 15 = \frac{1}{3} \times 1500 = \mathbf{500 \text{ cm}^3}$
4. $V = \frac{4}{3} \times 3.14 \times 125 = \frac{1570}{3} \approx \mathbf{523.3 \text{ cm}^3}$
5. Radius $= 3$ m. $V = \pi \times 9 \times 20 = 180\pi \approx \mathbf{565.5 \text{ m}^3}$

Summary

• Prisms and cylinders: $V = B \times h$ (base area × height).
• Pyramids and cones: $V = \dfrac{1}{3}Bh$ (one-third of the corresponding prism/cylinder).
• Sphere: $V = \dfrac{4}{3}\pi r^3$ — uses $r$ cubed, not squared.
• Always use radius (not diameter) in formulas involving $r$.
• Units are always cubic: cm³, m³, etc.

Related Topics

• Area and Perimeter — Formulas, Examples, and Practice
• Surface Area of 3D Shapes
• Area and Circumference of a Circle

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