Surface Area of a Cone
This worksheet practises using the formulae for the surface area of a cone.
You will need to calculate the curved surface area (πrl) and total surface area (πrl + πr²). You will also practise using Pythagoras to find the slant height, and working backwards from a given surface area to find missing lengths. Jump to the questions
Practise now
Worksheet preview and key skills
Worksheet preview
This worksheet generates questions on finding the curved and total surface area of a cone.
The progression moves from straightforward formula substitution towards using Pythagoras, and finishes with finding missing lengths from a known surface area.
What you’ll practise
- Calculating the curved surface area of a cone.
- Calculating the total surface area of a cone.
- Using Pythagoras to find the slant height when the perpendicular height is given.
- Working backwards to find the slant height or radius when the surface area is known.
- Giving exact answers in terms of π.
Use the interactive worksheet below, or read the Topic guide for the formulae and worked examples.
Surface area of a cone
Calculate the curved or total surface area of the cones, or work backwards to find a missing length.
Topic guide
The formulae
To find the surface area of a cone, you need to know the radius of the base (r) and the slant height (l).
Curved surface area = πrl
Total surface area = πrl + πr²
Radius, perpendicular height and slant height
It is important to use the correct measurements:
- Radius (r): The distance from the centre of the circular base to the edge. If you are given the diameter, halve it.
- Slant height (l): The distance along the sloping side of the cone. This is the length used in the surface area formulae.
- Perpendicular height (h): The straight vertical distance from the point (apex) to the centre of the base. Do not use this directly in the surface area formula.
Worked example 1: Total surface area
Find the total surface area of a cone with radius 5 cm and slant height 12 cm. Give your answer to 1 decimal place.
- Identify the measurements: r = 5, l = 12
- Calculate the curved surface: π × 5 × 12 = 60π
- Calculate the circular base: π × 5² = 25π
- Add them together: 60π + 25π = 85π
- Calculate the decimal: 85 × π ≈ 267.035...
- Round to 1 decimal place: 267.0 cm²
Worked example 2: Using Pythagoras
If you are given the perpendicular height (h) instead of the slant height (l), you must use Pythagoras' theorem first.
Find the total surface area of a cone with radius 3 cm and perpendicular height 4 cm.
- Use Pythagoras to find l: l² = 3² + 4²
- l² = 9 + 16 = 25
- l = √25 = 5 cm
- Now use the surface area formula: Total = π × 3 × 5 + π × 3² = 15π + 9π = 24π cm²
Reverse problems (working backwards)
You may be given the total surface area and asked to find the slant height or radius.
Example: The total surface area of a cone is 96π cm². Its radius is 6 cm. Find the slant height.
- Write the equation: πrl + πr² = 96π
- Substitute the radius (r = 6): 6πl + 36π = 96π
- Divide everything by π: 6l + 36 = 96
- Subtract 36: 6l = 60
- Divide by 6: l = 10 cm
Common mistakes
- Using the perpendicular height: Make sure you always use the slant height (l) in the formula. Use Pythagoras to find it if necessary.
- Adding the base incorrectly: Read the question carefully. Only add the base area (πr²) if it asks for the total surface area.
- Using the diameter: Always halve the diameter first to find the radius.
- Wrong units: Surface area is 2D, so it uses square units (e.g. cm², m²).