Take a rubber ball and drive a nail into it. Using the nail for support, wind a string tightly around the ball. When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered it; see the figure. Mark the starting and finishing points on the string, and unwind the string from the surface of the ball. Now, measure the diameter of the ball and get its radius. On a sheet of paper, draw four circles with a radius equal to the radius of the ball. Fill the circles with the string you had wound around the ball (as shown in the figure).
Answer: The string that covered the ball exactly fills four circles of the ball’s radius. So surface area of a sphere = 4 × πr2 = 4πr2.
Step-by-step solution
Idea: The same string covers the ball and then the circles, so the area it covers is the same both times.
- The string covers the whole curved surface of the ball, so the area it can cover = surface area of the ball.½ mark
- Unwound, the same string just fills four circles, each of radius r (the ball’s radius). Their area = 4 × πr2.1 mark
- So surface area of a sphere = 4πr2. For example, a ball of radius 3.5 cm: 4 × 227 × 12.25 = 154 cm2.½ mark
The string fills exactly four circles of radius r, so the surface area of a sphere is 4πr².
Check: Archimedes’ result: a sphere has the same area as the curved surface of the cylinder that just fits round it, 2πr × 2r = 4πr2 ✓.
Answer to write in the exam
String covering ball = string filling 4 circles of radius r
Area of 4 circles = 4πr2
∴ Surface area of sphere = 4πr2
Common mistakes that cost marks
- Using the diameter as the radius when drawing the circles; the circles then come out four times too big.
- Leaving gaps between turns of string, which makes it look as if fewer than four circles are filled.
How this can come in the exam
MCQ (1 mark)
The surface area of a sphere of radius r equals the total area of how many circles of radius r?
- 2
- 3
- 4
- 6
Show answer
(C) 4
4πr2 = 4 × πr2.
Try one yourself
The string from a ball fills four circles, each of area 38.5 cm2. Find the surface area and radius of the ball. (π = 227)
Show answer
Surface area = 4 × 38.5 = 154 cm2; πr2 = 38.5 ⇒ r2 = 12.25 ⇒ r = 3.5 cm.
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