Geometry & measurement · Grade 8
Volume of Cylinders, Cones and Spheres
Volume measures the space inside a three-dimensional object. These curved solids use formulas involving π and the radius.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Understand the key terms and formulas.
- Apply the method to worked and new examples.
- Check answers and explain the reasoning.
1. Radius and height
The radius r is the distance from the centre of a circular base to its edge. The height h is the perpendicular distance between bases, or from the apex to the base for a cone.
2. Cylinder
A cylinder has two equal circular bases. Its volume is base area × height: V = πr²h. Keep the radius squared and multiply by the perpendicular height.
3. Cone
A cone has one circular base and a vertex. Its volume is one-third of the matching cylinder: V = ⅓πr²h.
4. Sphere
A sphere is perfectly round in three dimensions. Its volume is V = ⁴⁄₃πr³. Remember that the radius is cubed.
5. Units and rounding
If measurements are in centimetres, volume is in cubic centimetres (cm³). Keep π in the calculation until the final step, then round as requested. Use the same units for all lengths.
Compare solid volumes and use units
Volume measures the space inside a three-dimensional solid and is written in cubic units. A cylinder has volume V = πr²h. A cone with the same base radius and height has one-third of the cylinder's volume: V = ⅓πr²h. A sphere has volume V = ⁴⁄₃πr³. In each formula, r is the radius, not the diameter.
Worked reasoning: A cone has radius 3 cm and height 4 cm. Its volume is ⅓ × π × 3² × 4 = 12π cm³, approximately 37.7 cm³. If the question gives a diameter of 6 cm, first divide by 2 to get the radius 3 cm.
Keep π in the exact answer when appropriate, then round the decimal only at the end. The final unit must be cubic, such as cm³ or m³, because three dimensions are multiplied.
- Do not use diameter in place of radius.
- Square the radius for cylinder and cone formulas; cube it for a sphere.
- Convert all measurements to the same unit before calculating.
Worked examples
- Cylinder r = 3 cm, h = 5 cm: V = π(3²)(5) = 45π ≈ 141.4 cm³.
- Cone r = 3 cm, h = 5 cm: V = ⅓π(9)(5) = 15π ≈ 47.1 cm³.
- Sphere r = 3 cm: V = ⁴⁄₃π(27) = 36π ≈ 113.1 cm³.
Independent practice
Attempt each problem before reviewing the worked answer.
- Find cylinder volume for r = 2 cm, h = 7 cm. Answer: 28π ≈ 88.0 cm³.
- Find cone volume for r = 3 cm, h = 8 cm. Answer: 24π ≈ 75.4 cm³.
- Find sphere volume for r = 2 cm. Answer: 32π/3 ≈ 33.5 cm³.
- A cylinder has diameter 10 cm and height 4 cm. Answer: r = 5; V = 100π ≈ 314.2 cm³.
- What happens to cylinder volume if height doubles? Answer: It doubles.
- What happens to sphere volume if radius doubles? Answer: It becomes 8 times as large.
- A cone and cylinder have the same base and height. Answer: Cone volume is one-third of cylinder volume.
- Why is volume written in cubic units? Answer: It measures three-dimensional space (length × width × height).
Knowledge check
Choose one answer for each question, then check for explanations.
Printable worksheet and answer key
Use the independent-practice questions above as a worksheet. Show each step, include units where appropriate, and check your answers after attempting the full set.
Study reminder
Write down the formula or reasoning used for each answer.