But you may be able to find your own derivation of the formula by connecting it with the formula for volume of a cone; namely, by writing it as
V = 13 × (4πr2) × r,
i.e., Volume of sphere = 13 × surface area of sphere × radius of sphere.
Try to work out the connecting links on your own.
Answer: Split the sphere’s surface into tiny patches and join each to the centre: the sphere becomes many thin cones, all of height r. Their volumes add to r3 × (total base area) = r3 × 4πr2 = 43πr3.
Step-by-step solution
Idea: A cone’s volume is 13 × base area × height. If many thin cones share the same height r, their total volume is r3 × (sum of their base areas), and those bases cover the whole sphere.
- Divide the whole surface of the sphere into very tiny regions. Their shapes do not matter; they only need to be very small.½ mark
- Join every point of each tiny region to the centre O. This cuts the sphere into many thin cone-like pieces. Each has its apex at O and height r (the distance from O to the surface).1 mark
- Volume of one thin cone = 13 × (its base area) × r. Adding all of them: volume of sphere = r3 × (sum of all base areas) = r3 × surface area of sphere.1 mark
- Surface area = 4πr2, so volume = r3 × 4πr2 = 43πr3.½ mark
Volume of sphere = (r/3) × (sum of the bases of the thin cones) = (r/3) × 4πr² = (4/3)πr³.
Check: For r = 3: 13 × 4π × 9 × 3 = 36π and 43π × 27 = 36π ✓.
Answer to write in the exam
Sphere = many thin cones, apex at centre, height r
Volume of each = 13 × base area × r
Volume of sphere = r3 × (sum of base areas) = r3 × 4πr2
∴ Volume of sphere = 43πr3
Common mistakes that cost marks
- Taking the height of each small cone as the diameter 2r. Each cone runs from the centre to the surface, so its height is r.
- Forgetting that the 13 comes from the cone formula, and writing volume = surface area × radius.
How this can come in the exam
MCQ (1 mark)
For a sphere, volumesurface area equals
- r
- r3
- 3r
- r4
Show answer
(B) r3
43πr3 ÷ 4πr2 = r3.
Short answer (2 marks)
The surface area of a sphere is 616 cm2. Using volume = 13 × surface area × radius, find its volume. (π = 227)
Show answer
4 × 227 × r2 = 616 ⇒ r2 = 49 ⇒ r = 7 cm. Volume = 13 × 616 × 7 = 43123 ≈ 1437.33 cm3.Try one yourself
A sphere has surface area 36π cm2. Find its volume using the cone idea.
Show answer
4πr2 = 36π ⇒ r = 3 cm; volume = 13 × 36π × 3 = 36π cm3.
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