Sheet № 170 · Higher only · AQA · Edexcel · OCR
Surface Area of a Sphere –
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.
§Key definitions
Question:
A sphere has a radius of 7 cm. Find the surface area to 1 decimal place.
Answer:
615.8 cm²
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.
§Formulas to memorise
Surface area of a sphere = 4πr²
Curved surface area of a hemisphere = 2πr²
Total surface area of a hemisphere = 2πr² + πr² = 3πr²
For a sphere, substitute into SA = 4πr².
Worked example
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.)
⚠ 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.