Solids: Volume, Sections & Rotations — Print
Every solid here is a flat shape that has been stacked, tapered or spun.
What the practice 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.
Take a rectangle and spin it about one of its edges. What does it sweep out?
The edge stays put. The far side traces a circle. Every point in between traces a circle too.
The result is a cylinder — and its radius is the rectangle's width, its height the rectangle's height.
Spin a right triangle about a leg instead and you get a cone. Spin a semicircle about its diameter and you get a sphere.
Three solids, three flat shapes, one idea.
And it runs the other way too. Slice a solid and you get a flat shape back — which is why cross sections and solids of revolution are two halves of the same unit.
Stacked solids: prisms and cylinders
A prism is a base shape stacked straight up. Its volume is the area of the base times the height: V = Bh.
A cylinder is the same idea with a circular base, so V = π r2 h — it is not a separate formula, just Bh with B = π r2.
Because of Cavalieri, this works even if the stack leans over. Only the perpendicular height counts.
One formula, many solids
V = Bh covers every prism and cylinder. Find the area of the base and multiply by the height — the shape of the base is the only thing that changes.
Tapered solids: pyramids and cones
Taper a prism to a point and you get a pyramid, which holds exactly one third as much: V = 13 Bh.
A cone is a pyramid with a circular base, so V = 13 π r2 h.
The 13 is the same one you can pour — three coneloads fill the matching cylinder.
Spheres
V = 43 π r3 and surface area = 4π r2.
The surface area is worth noticing: it is exactly four times the area of the circle you would get by slicing through the middle.
A sphere holds two thirds of the cylinder that just contains it — Archimedes' result, and the one on his tomb.
Cross sections
Slicing a solid exposes a flat shape, and which shape depends on the angle of the cut.
A cut parallel to a cylinder's base gives a circle; a vertical cut through the axis gives a rectangle; a slanted cut gives an ellipse.
A cube can be sliced to give a triangle, a rectangle, a pentagon or a regular hexagon — the solid never changed, only the plane.
Surface area
Surface area is the total of every face. The reliable method is to unfold the solid into its net and add up the flat pieces.
A cylinder unrolls into two circles and one rectangle whose width is the circumference: SA = 2π r2 + 2π r h.
Volume is in cubic units and surface area in square units, so the units on your answer say which one you found.
Slant height, not vertical height
The lateral surface of a cone uses the slant height ℓ, not the vertical height h. They are different, and related by ℓ2 = r2 + h2.
Worked examples
Prism volume
A prism has a triangular base with legs 6 and 8, and a height of 15. Find its volume.
- Base area first: B = 12 (6)(8) = 24 A right triangle.
- V = Bh = 24 × 15 Stacked solid, so base times height.
- V = 360 cubic units Cubic units, because it fills space.
360 cubic units
Cone surface area
A cone has radius 5 and vertical height 12. Find its total surface area in terms of π.
- Slant height: ℓ = √52 + 122 = 13 Lateral area needs ℓ, not h.
- Lateral area = π r ℓ = π(5)(13) = 65π The curved surface.
- Base = π r2 = 25π The circle underneath.
- Total = 65π + 25π = 90π A 5-12-13 triangle made the slant height clean ✓
90π square units
A solid of revolution
A right triangle with legs 3 and 4 is spun about the leg of length 4. Describe the solid and find its volume in terms of π.
- Spinning a right triangle about a leg gives a cone. The axis leg becomes the height.
- The axis is the leg of 4, so h = 4; the other leg sweeps the base, so r = 3. Which leg you spin about decides which is which.
- V = 13 π (3)2 (4) = 13 π (36) Substitute.
- V = 12π Spinning about the other leg would give 16π — a different cone.
A cone; 12π
Words to know
- Prism
- Two congruent parallel bases joined by rectangles. V = Bh.
- Pyramid
- A base tapering to a point. V = 13 Bh.
- Cross section
- The flat face exposed by slicing a solid.
- Solid of revolution
- The solid swept out by spinning a flat shape about a line.
- Lateral area
- The surface area not counting the bases.
- Cavalieri's principle
- Same height and same cross sections at every level means same volume.
Mistakes to watch for
Using the vertical height where the slant height belongs. A cone's lateral area needs ℓ, and ℓ = √r2 + h2 is always longer than h. Using h makes the surface come out too small.
Forgetting the 13 on pyramids and cones. Tapered solids hold a third of the prism or cylinder that wraps them — you can pour it three times to check, and without the third the answer is triple.
Mixing volume and surface area. Volume fills and is measured in cubic units; surface area wraps and is measured in square units. The units on your answer identify which one you actually computed.
Assuming every cross section matches the base. A slanted cut through a cylinder gives an ellipse, not a circle, and a cube can be cut into a hexagon. The angle of the plane decides the shape, not the solid alone.
Cavalieri's principle, from 1635, says two solids with the same height and the same cross-sectional area at every level have the same volume.
So a leaning stack of coins holds exactly as much as a straight one — which is why an oblique prism uses the same formula as a right one.
It was one of the ideas that led directly to integral calculus fifty years later.
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