Volume of Cylinders, Cones & Spheres — Challenge
Three formulas, one family. A cone is a third of its cylinder, and a sphere is two thirds of it.
What these problems cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Volume — Space inside, in cubic units.
- Factor — A number that divides evenly into another.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Cylinder — Two parallel circles joined by a curved surface. V = π r2 h.
- Cone — A circle tapering to a point. V = 13π r2 h.
- Sphere — A ball. V = 43π r3.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
What is the volume of a cylinder with radius 2 and height 5?
- 10π
- 20π
- 40π
- 100π
- 4π
Worked answer
20π
The volume of a cylinder is π r2h.
The base area is π· 22 = 4π, and multiplying by the height 5 gives 20π.
Problem 2
What is the volume of a cone with radius 3 and height 4?
- 9π
- 12π
- 36π
- 4π
- 108π
Worked answer
12π
The volume of a cone is one third of π r2h.
π· 32· 4 = 36π would be the cylinder.
A third of that is 12π.
Problem 3
What is the volume of a sphere of radius 6?
- 144π
- 216π
- 288π
- 864π
- 72π
Worked answer
288π
The volume of a sphere is 43π r3.
63 = 216, so the volume is 43π· 216.
43· 216 = 288, giving 288π.
Problem 4
A cylinder has a volume of 48π and a height of 3. What is its radius?
- 2
- 4
- 8
- 12
- 16
Worked answer
4
From π r2h = 48π with h=3, divide both sides by 3π to get r2 = 16.
Taking the positive square root gives r = 4.
Stopping at 16 gives the squared radius rather than the radius.
Problem 5
What is the volume of a cube with side 5?
- 25
- 75
- 100
- 125
- 150
Worked answer
125
The volume of a cube is the side cubed.
53 = 125 cubic units. The value 25 is the area of one face.
Problem 6
A cylinder has a volume of 60. A cone with the same base and height has what volume?
- 15
- 20
- 30
- 40
- 180
Worked answer
20
A cone holds exactly one third of the cylinder that shares its base and height.
603 = 20 cubic units.
Halving would give 30, which is the wrong fraction.
Problem 7
A sphere has a volume of 36π. What is its radius?
- 2
- 3
- 4
- 6
- 9
Worked answer
3
From 43π r3 = 36π, divide both sides by π and multiply by 34 to get r3 = 27.
The cube root of 27 is 3, so the radius is 3.
Problem 8
What is the volume of a cylinder with radius 1 and height 10?
- π
- 5π
- 10π
- 20π
- 100π
Worked answer
10π
The volume is π r2h, and r=1 makes r2 = 1.
So the volume is π· 1· 10 = 10π.
Squaring the height instead would give 100π, which is the trap among the options.