Solids: Volume, Sections & Rotations — Practice
Every solid here is a flat shape that has been stacked, tapered or spun.
What these questions cover
These questions use 16 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Cross section — The 2-D face revealed when a plane cuts through a solid.
- Volume — Space inside, in cubic units.
- Surface area — Total area of all the outside faces, in square units.
- Prism — A solid with two identical parallel bases joined by rectangles.
- Base — The number being multiplied. In 53 the base is 5.
- Factor — A number that divides evenly into another.
- Radius — Centre to edge. Half the diameter.
- Scale — What one grid step is worth — and where the axis starts.
- 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.
- Parallel — Same slope, different intercept.
- Pyramid — A base tapering to a point. V = 13 Bh.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
A prism has base area 24 and height 15. Find its volume.
Worked answer
360
24 × 15 = 360
Problem 2
Find the volume of a cylinder with r = 4 and h = 9, in terms of π.
Worked answer
144pi
π(16)(9) = 144π
Problem 3
A pyramid has base area 30 and height 12. Find its volume.
Worked answer
120
13 (30)(12) = 120
Problem 4
Find the volume of a sphere with r = 3, in terms of π.
Worked answer
36pi
43 π (27) = 36π
Problem 5
A cut parallel to the base of a cylinder gives what shape?
Worked answer
circle
The cross section matches the base: a circle.
Problem 6
Find the volume of a cone with r = 6 and h = 10, in terms of π.
Worked answer
120pi
13 π (36)(10) = 120π
Problem 7
A cone has r = 5 and h = 12. Find its slant height.
Worked answer
13
√25 + 144 = √169 = 13
Problem 8
A cone has r = 5 and ℓ = 13. Find its total surface area in terms of π.
Worked answer
90pi
Lateral = π r ℓ = 65π
Base = 25π
Total = 90π
Problem 9
Find the surface area of a sphere with r = 6, in terms of π.
Worked answer
144pi
4π(36) = 144π
Problem 10
A rectangle is spun about one of its sides. What solid is formed?
Worked answer
cylinder
The result is a cylinder.
Problem 11
A right triangle is spun about one of its legs. What solid is formed?
Worked answer
cone
The result is a cone.
Problem 12
A slanted cut through a cylinder gives
- an ellipse
- a circle
- a rectangle
- a triangle
Worked answer
an ellipse
Parallel to the base gives a circle.
A slant stretches it into an ellipse.
Problem 13
A right triangle with legs 3 and 4 is spun about the leg of length 4. Find the volume in terms of π.
Worked answer
12pi
r = 3, h = 4
13 π (9)(4) = 12π
Problem 14
A cylinder of height 4 has volume 100π. Find its radius.
Worked answer
5
π r2(4) = 100π
r2 = 25
r = 5
Problem 15
A silo is a cylinder r = 3, h = 10 topped by a cone of the same radius and height 4. Find the total volume in terms of π.
Worked answer
102pi
Cylinder 90π
Cone 12π
Total 102π
Problem 16
A solid is scaled up by a factor of 3. By what factor does its volume grow?
Worked answer
27
33 = 27