Volumes (3D Mensuration)

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Quick recall

• Volume uses cubic units; surface area uses square units.
• Convert lengths to one unit before multiplying.

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Formula Cards

1. Cube volume

$$V=s^3$$

2. Cube surface area

$$SA=6s^2$$

3. Cuboid volume

$$V=lwh$$

4. Cuboid surface area

$$SA=2(lw+wh+hl)$$

5. Cylinder volume

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

6. Cylinder curved surface area

$$CSA=2\pi rh$$

7. Sphere volume

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

8. Sphere surface area

$$SA=4\pi r^2$$

9. Cone volume

$$V=\frac{1}{3}\pi r^2h$$

Worked Examples

Example 1

Question: A cylinder has radius 3 units and height 10 units. Find its volume.

Step 1: Use the cylinder formula.

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

Step 2: Substitute.

$$V=\pi(3^2)(10)=90\pi$$

Answer: 90π cubic units.

Example 2

Question: Find the volume of a cube with side 5 units.

Step 1: Cube the side length.

$$V=5^3$$

Step 2: Calculate.

$$V=125$$

Answer: 125 cubic units.

Example 3

Question: Find the volume of a sphere with radius 6 units.

Step 1: Cube the radius.

$$6^3=216$$

Step 2: Multiply by 4π/3.

$$V=\frac{4}{3}\pi(216)=288\pi$$

Answer: 288π cubic units.

Example 4

Question: Find the volume of a cone with radius 3 units and height 7 units.

Step 1: Apply one-third of the cylinder formula.

$$V=\frac{1}{3}\pi r^2h$$

Step 2: Substitute.

$$V=\frac{1}{3}\pi(9)(7)=21\pi$$

Answer: 21π cubic units.

Drill

Try each question before opening its answer.

Question 1

Volume of a cuboid 4×5×6.

• Show answer

  120

Question 2

Volume of a cube with side 4.

• Show answer

  64

Question 3

Volume of a cylinder r=7, h=10 (π=22/7).

• Show answer

  1540

Question 4

Volume of a sphere with r=3.

• Show answer

  36π ≈ 113.1

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