Surface Area of Spheres and Cones: GCSE Maths

The surface area of a sphere uses SA = 4πr², and the total surface area of a cone uses SA = πrl + πr², where l is the slant height. Both formulae appear on the GCSE formula sheet. The key challenge is using the slant height l rather than the vertical height h.

What is the surface area of a sphere?

A sphere has no flat faces, edges, or corners — it is entirely curved. Its surface area is found using a single formula:

SA = 4πr²

where r is the radius of the sphere. This formula gives the total surface area — there is no distinction between "curved" and "total" for a sphere since the entire surface is curved.

The factor of 4 is not arbitrary: the surface area of a sphere is exactly four times the area of a great circle (a circle with the same radius as the sphere). The formula for the area of a circle is πr², and 4 × πr² gives the surface area of the sphere.

Quantity | Formula
Area of a circle with radius r | πr²
Surface area of a sphere with radius r | 4πr²

How do you calculate the surface area of a sphere?

Worked example 1: Find the surface area of a sphere with radius 5 cm. Give your answer in terms of π and as a decimal to 1 decimal place.

Write the formula: SA = 4πr².

Substitute r = 5: SA = 4π(5²) = 4π × 25.

Simplify: SA = 100π cm² (exact answer).

Decimal: SA = 100 × 3.14159… ≈ 314.2 cm².

Worked example 2: A sphere has surface area 144π cm². Find its radius.

Set up the equation: 4πr² = 144π.

Divide both sides by 4π: r² = 36.

Take the square root: r = 6 cm.

When working backwards from a surface area to find r, always divide by 4π first to isolate r², then square root.

What is the curved surface area of a cone?

A cone has two parts: a circular base and a curved lateral surface. The curved surface area covers only the sloping sides, not the base.

Curved SA = πrl

where r is the radius of the base and l is the slant height — the distance measured along the sloping side from the base edge to the apex.

The slant height l is NOT the same as the vertical height h. Confusing them is the single most common error in this topic. If the question gives you h instead of l, you must calculate l first using Pythagoras' theorem:

l = √(r² + h²)

This is because the slant height, radius, and vertical height form a right-angled triangle, with l as the hypotenuse.

How do you calculate the total surface area of a cone?

The total surface area of a cone is the sum of the curved surface and the circular base:

Total SA = πrl + πr²

This can also be written as πr(l + r) after factorising out πr.

Worked example 3: A cone has radius 3 cm and slant height 7 cm. Find the total surface area. Leave your answer in terms of π.

Curved surface area = πrl = π × 3 × 7 = 21π cm².

Base area = πr² = π × 3² = 9π cm².

Total SA = 21π + 9π = 30π cm².

Worked example 4: A cone has radius 4 cm and vertical height 3 cm. Find the total surface area to 1 decimal place.

Find the slant height first: l = √(r² + h²) = √(4² + 3²) = √(16 + 9) = √25 = 5 cm.

Curved SA = π × 4 × 5 = 20π.

Base area = π × 4² = 16π.

Total SA = 36π ≈ 113.1 cm².

How do you find the slant height when it is not given?

If a question provides the vertical height h and the base radius r, use Pythagoras' theorem:

l = √(r² + h²)

r (cm) | h (cm) | Calculation | l (cm)
3 | 4 | √(9 + 16) = √25 | 5
5 | 12 | √(25 + 144) = √169 | 13
6 | 8 | √(36 + 64) = √100 | 10
2 | 2 | √(4 + 4) = √8 | 2√2 ≈ 2.83

Always draw and label a cross-section of the cone showing r, h, and l before substituting any values. This makes the right-angled triangle visible and reminds you which measurement is which.

What are the most common mistakes?

Mistake | Consequence | Correct approach
Using h instead of l in the curved SA formula | Incorrect answer — often significantly wrong | Always find l first using l = √(r² + h²)
Forgetting the base when finding total SA | Only partial credit for the calculation | Total SA = curved SA + πr²
Using diameter instead of radius | Answer is 4× too large | Halve the diameter before substituting
Rounding l before substituting | Introduces rounding error in the final answer | Keep l as an exact value until the last step

Frequently asked questions

Why does the cone formula use slant height rather than vertical height?

The curved surface of a cone, when "unrolled" flat, forms a sector of a circle. The radius of that sector is the slant height l — the actual distance from the apex to the base edge, measured along the surface. The vertical height h goes straight down inside the cone and does not touch the curved surface, so it is irrelevant to the surface area calculation. Pythagoras connects all three: l² = r² + h².

Can the surface area of a sphere be a whole number?

Only if the radius makes 4πr² a whole number or a simple expression. In practice, surface area answers almost always involve π or its decimal approximation. Answers in terms of π (such as 100π) are exact; decimal answers are approximate. GCSE questions often specify which form to use — always read the instruction before rounding.

What is the difference between total surface area and curved surface area for a cone?

The curved surface area (CSA = πrl) covers only the sloping lateral surface. The total surface area (TSA = πrl + πr²) includes both the curved surface and the circular base. If a question asks for the "total surface area," include the base. If it asks only for the "curved surface area," leave the base out. Many questions distinguish clearly between the two, but if "surface area" is stated without a qualifier, assume total.

Does a sphere have a volume formula too?

Yes: volume of a sphere = (4/3)πr³. Surface area and volume are related — both use r — but they are different calculations. Surface area measures the outer covering (in cm² or m²); volume measures the interior space (in cm³ or m³). At GCSE Higher, questions sometimes combine both: for example, finding the radius from the surface area, then using it to calculate the volume, or comparing two similar spheres by their surface area and volume scale factors.

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