The surface area of a sphere is a Higher tier formula that appears frequently in GCSE exams. Questions often extend to hemispheres, where you must remember to include the flat circular face as well as the curved surface.
What Is the Surface Area of a Sphere?
The surface area of a sphere is the total area of its outer curved surface. Unlike a cube or cuboid, a sphere has no edges or flat faces — every point on its surface is the same distance (the radius) from the centre.
The formula SA = 4πr² tells us that the surface area of a sphere is exactly four times the area of a circle with the same radius. This elegant relationship was first proved by Archimedes.
For a hemisphere (half a sphere), the curved surface area is 2πr² (half of 4πr²), but the total surface area also includes the flat circular base of area πr², giving a total of 3πr².
Key Formulas
Step-by-Step Method
- Identify the radius. If the diameter is given, halve it.
- For a sphere, substitute into SA = 4πr².
- For a hemisphere, calculate the curved surface (2πr²) and add the flat circular base (πr²) unless the question specifies curved surface only.
Worked Example 1 — Foundation Level
Question: A sphere has a radius of 7 cm. Find the surface area to 1 decimal place.
Working: SA = 4πr² SA = 4 × π × 7² SA = 4 × π × 49 SA = 196π SA = 615.8 (1 d.p.)
Answer: 615.8 cm²
Worked Example 2 — Higher Level
Question: A hemisphere has a radius of 6 cm. Calculate the total surface area in terms of π.
Working: Curved surface = 2πr² = 2π × 36 = 72π Flat base = πr² = π × 36 = 36π Total SA = 72π + 36π = 108π
Answer: 108π cm²
Worked Example 3 — Exam Style
Question: The surface area of a sphere is 400π cm². Find the radius.
Working: SA = 4πr² 400π = 4πr² Divide both sides by 4π: r² = 100 r = √100 r = 10
Answer: 10 cm
Common Mistakes
- Forgetting the flat face of a hemisphere. A hemisphere's total surface area is 3πr², not 2πr². The 2πr² is only the curved part — you must add the circular base πr².
- Using diameter instead of radius. The formula uses the radius. If the question gives a diameter of 10 cm, use r = 5 cm.
- Squaring incorrectly. Remember that r² means r × r, not 2r. For r = 7, r² = 49, not 14.
- Confusing surface area with volume. Surface area is in square units (cm²); volume is in cubic units (cm³). The sphere volume formula is (4/3)πr³ — do not mix them up.
Exam Tips
- The formula SA = 4πr² is given on the exam formula sheet, but practise using it so you do not waste time looking it up.
- For hemisphere questions, always ask yourself: "Do they want curved surface area or total surface area?"
- If asked to find the radius from a given surface area, rearrange: r² = SA ÷ (4π), then square root.
Practice Questions
Q1 (Foundation): A sphere has a radius of 3 cm. Find its surface area to 1 decimal place.
Q2 (Foundation): A sphere has a diameter of 16 cm. Find its surface area in terms of π.
Q3 (Higher): A solid hemisphere has a radius of 5 cm. Find the total surface area to 1 decimal place.
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Related Topics
Summary
- Surface area of a sphere = 4πr².
- A sphere's surface area is exactly four times the area of a great circle.
- For a hemisphere, the total surface area = 3πr² (curved 2πr² plus flat base πr²).
- Always check whether radius or diameter is given and convert if needed.
- To find the radius from surface area, rearrange: r = √(SA ÷ 4π).
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.
Cambridge problems on area, circumference, arcs and sectors.
University of Cambridge · Free · Open AccessArea formulas, circle calculations, sectors and segments.
Corbett Maths · Free · Open AccessVolume and surface area explorations from Cambridge.
University of Cambridge · Free · Open AccessVolume of prisms, cylinders, cones, spheres and compound shapes.
Corbett Maths · Free · Open Access