Volume of Cylinders, Cones & Spheres — Practice
Three formulas, one family. A cone is a third of its cylinder, and a sphere is two thirds of it.
What these questions 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
Find the volume of a cylinder with r = 3 and h = 5, in terms of π.
Worked answer
45pi
π(9)(5)
= 45π
Problem 2
Find the volume of a cylinder with r = 2 and h = 10, in terms of π.
Worked answer
40pi
π(4)(10)
= 40π
Problem 3
Find the volume of a cone with r = 3 and h = 4, in terms of π.
Worked answer
12pi
13π(9)(4)
= 12π
Problem 4
Find the volume of a sphere with r = 3, in terms of π.
Worked answer
36pi
33 = 27
43π(27) = 36π
Problem 5
A cylinder has diameter 14 cm. What radius goes into the formula?
Worked answer
7
14 ÷ 2 = 7 cm
Problem 6
Find the volume of a cylinder with diameter 10 and height 6, in terms of π.
Worked answer
150pi
r = 5
π(25)(6) = 150π
Problem 7
Find the volume of a cone with r = 6 and h = 10, in terms of π.
Worked answer
120pi
13π(36)(10)
= 360π3 = 120π
Problem 8
Find the volume of a sphere with diameter 6, in terms of π.
Worked answer
36pi
r = 3
43π(27) = 36π
Problem 9
Find the volume of a cylinder with r = 4 and h = 7, rounded to one decimal place. Use π ≈ 3.14159.
Worked answer
351.9
π(16)(7) = 112π
112 × 3.14159 ≈ 351.9
Problem 10
A cylinder of height 4 has volume 100π. Find its radius.
Worked answer
5
π r2 (4) = 100π
r2 = 25
r = 5
Problem 11
A cone and a cylinder have the same radius and height. The cone's volume is
- one third of the cylinder's
- the same
- twice as much
- three times as much
Worked answer
one third of the cylinder's
Cylinder: π r2 h
Cone: 13π r2 h
So one third
Problem 12
A cone with radius 3 has volume 30π. Find its height.
Worked answer
10
13π(9)h = 30π
3h = 30
h = 10
Problem 13
A silo is a cylinder of radius 3 and height 10 topped by a cone of the same radius and height 4. Find the total volume in terms of π.
Worked answer
102pi
Cylinder: π(9)(10) = 90π
Cone: 13π(9)(4) = 12π
90π + 12π = 102π
Problem 14
A sphere has volume 256π3. Find its radius.
Worked answer
4
43r3 = 2563
4r3 = 256
r3 = 64, so r = 4
Problem 15
A cylindrical glass of radius 4 and height 12 is filled, then a sphere of radius 3 is dropped in. How much water in terms of π overflows?
Worked answer
36pi
Sphere: 43π(27) = 36π
The glass was already full
So 36π overflows
Problem 16
A cylinder's radius is doubled and its height stays the same. By what factor does the volume grow?
Worked answer
4
V = π r2 h
Doubling r gives (2r)2 = 4r2
So the volume is 4 times as large