Show that the volume of a sphere is equal to 23 of the volume of the smallest cylinder which encloses it.
Step-by-step solution
Idea: The tightest cylinder around a sphere touches it all round and at the top and bottom, so its radius is r and its height is the diameter 2r.
- Smallest enclosing cylinder: radius = r, height = 2r.1 mark
- Volume of cylinder = πr2 × 2r = 2πr3. Volume of sphere = 43πr3.1 mark
- volume of spherevolume of cylinder = (4/3)πr32πr3 = 46 = 23. So the sphere is 23 of the cylinder. Shown.1 mark
Check: r = 3: cylinder 54π, sphere 36π, and 36 ÷ 54 = 23 ✓.
Answer to write in the exam
Cylinder: radius r, height 2r
Vcyl = πr2(2r) = 2πr3
Vsphere = 43πr3 = 23 × 2πr3
∴ Volume of sphere = 23 × volume of cylinder
Common mistakes that cost marks
- Taking the cylinder’s height as r instead of 2r.
- Taking the cylinder’s radius as 2r (the diameter).
How this can come in the exam
Assertion (A): The curved surface area of the smallest cylinder around a sphere equals the surface area of the sphere.
Reason (R): The volume of a sphere is 23 of the volume of that cylinder.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(B) Both A and R are true, but R is not the correct explanation of A.
A: 2πr × 2r = 4πr2, true. R is true too, but a volume fact does not explain an area fact.
Try one yourself
A ball of radius 7 cm just fits in a cylindrical tin. Find the empty space in the tin. (π = 227)
Show answer
Tin = 227 × 49 × 14 = 2156; ball = 23 × 2156 ≈ 1437.33; empty = 13 × 2156 ≈ 718.67 cm3.
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