Surface area (AQA GCSE Maths): Revision Notes

Surface area

Surface area calculations are essential for finding how much material is needed to cover 3D shapes. Different shapes have different formulas, and it's crucial to understand when and how to apply each one.

Cone surface area

A cone has two types of surface area that you need to understand.

Curved surface area of a cone

The curved surface area only includes the slanted side of the cone, not the circular base. The formula is:

$\text{Curved surface area of cone} = \pi r l$

Where:

• $r$ = radius of the base
• $l$ = slant height (not the vertical height!)

Total surface area of a cone

To find the complete surface area, you must add the area of the circular base:

$\text{Total surface area of cone} = \pi r^2 + \pi r l$

This combines the base area ($\pi r^2$) with the curved surface area ($\pi r l$).

Finding the slant height

When you're given the vertical height and radius, you can find the slant height using Pythagoras' theorem.

Using Pythagoras:

$l^2 = h^2 + r^2$

Where:

• $l$ = slant height
• $h$ = vertical height
• $r$ = radius

Sphere surface area

The surface area of a sphere uses this formula:

$\text{Surface area of sphere} = 4\pi r^2$

Hemisphere surface area

A hemisphere is exactly half a sphere. The curved surface area is:

$\text{Curved surface area of hemisphere} = \frac{1}{2} \times 4\pi r^2 = 2\pi r^2$

Compound shapes

Complex 3D shapes can often be broken down into simpler parts. The key strategy is:

For compound shapes made from cylinders and hemispheres, break them into individual components and calculate each surface area before combining the results.

Exam tips

Practice problems

When tackling surface area problems, applying the systematic approach will lead to accurate and complete solutions.