Volume & Surface Area of 3-D Figures — Print
Volume fills the inside — base area times height. Surface area wraps the outside — add up every face.
What the practice questions cover
These questions use 15 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.
- 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.
- Radius — Centre to edge. Half the diameter.
- Cylinder — Two parallel circles joined by a curved surface. V = π r2 h.
- Cone — A circle tapering to a point. V = 13π r2 h.
- Parallel — Same slope, different intercept.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Height — The perpendicular distance from the base to the opposite corner.
- Rectangular prism — A box — six rectangular faces meeting at right angles.
- Edge length — The measurement along one side.
Take a cereal box and cut it open along the edges, then lay it flat.
What you now have is six rectangles — the front, the back, two sides, a top and a base.
Add up those six areas and you have the amount of card the box is made from. That is the surface area.
But the cereal inside was never measured by that. The cereal filled the space, and space needs three measurements: length, width and height.
One is about the skin, the other about the filling — and they even have different units.
Surface area is measured in cm2 because it is a flat thing made of lengths times lengths. Volume is cm3. If the units on your answer are wrong, you have computed the other quantity.
One formula for every prism and cylinder
V = Bh, where B is the area of the base and h is the height.
Think of it as stacking: one layer of the base is B cubic units thick, and you stack h of them.
Rectangular prism: B = lw, so V = lwh. Triangular prism: B = 12bh△. Cylinder: B = π r2, so V = π r2 h. Same idea each time — only B changes.
Find the base first
The base is the shape that repeats all the way through the solid. Find its area, then multiply by how far it repeats. That single habit handles every prism.
Surface area is just adding faces
Unfold the solid into its net, find each piece's area, and add.
Rectangular prism: three pairs of matching faces, so SA = 2lw + 2lh + 2wh.
Cylinder: two circles plus one rectangle. The rectangle is the label peeled off — its width is the circumference. So SA = 2π r2 + 2π r h.
Keep the units straight
Volume is in cubic units: cm3, m3, in3. Surface area is in square units: cm2, m2.
If your answer has the wrong kind of unit, you used the wrong formula. This is a free check on every problem.
Ask "filling or wrapping?"
Filling a fish tank with water, or a box with sand → volume. Wrapping a present, or painting a wall → surface area. Saying which one out loud before you start prevents most errors in this unit.
Pyramids and cones hold a third
A pyramid or cone with the same base and height as a prism or cylinder holds exactly one third as much: V = 13Bh.
You can see this by pouring: three cone-fulls of water fill a cylinder of matching base and height exactly.
Working backwards
Given the volume and two dimensions, divide to find the third. If V = 240 cm3 with base 8 × 5, then 240 = 40h, so h = 6 cm.
Given surface area, substitute and solve — often an equation from Unit 11 appears inside a geometry problem.
Worked examples
Triangular prism
A prism has a triangular base with base 6 cm and height 4 cm, and the prism is 10 cm long. Find its volume.
- Find B, the triangle: 12(6)(4) = 12 cm2 The triangle is the shape that repeats.
- The prism length is the height of the solid: 10 cm. How far the triangle repeats.
- V = Bh = 12 × 10 = 120 cm3 Base area times height.
120 cm3.
Cylinder, both measures
A can has radius 3 cm and height 8 cm. Find its volume and surface area. Use π ≈ 3.14.
- Volume: π r2 h = π (9)(8) = 72π ≈ 226.1 cm3 Square the radius first, then multiply.
- Two circles: 2π r2 = 2π(9) = 18π Top and bottom.
- The label: 2π r h = 2π(3)(8) = 48π A rectangle, circumference wide and h tall.
- Total: 18π + 48π = 66π ≈ 207.2 cm2 Add all the faces.
V ≈ 226.1 cm3; SA ≈ 207.2 cm2.
Backwards from volume
A rectangular tank holds 1,680 cm3. Its base is 14 cm by 10 cm. How deep is the water when full?
- Base area: 14 × 10 = 140 cm2 That is B.
- 1680 = 140h Put the known volume into V = Bh.
- h = 1680 ÷ 140 = 12 Divide both sides by 140.
12 cm deep.
Words to know
- 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.
- Lateral area
- The side faces only, not counting the two bases.
Mistakes to watch for
Mixing up volume and surface area. Volume fills, surface area wraps. The units tell you instantly which you computed: cm3 fills a space, cm2 covers one.
Using the diameter in π r2. It must be halved first — and because r is squared, forgetting makes the answer four times too big rather than twice.
Forgetting a face in surface area. A box has six, not four: the top and base are easy to lose because you cannot see both at once in a drawing. Unfold it and count.
Doubling the volume when you double a side. Doubling every edge multiplies the volume by eight, since all three dimensions grew. A box twice as tall only doubles it.
Hiero of Syracuse suspected his goldsmith had mixed silver into a gold crown. Archimedes is said to have solved it in the bath, noticing the water rise.
Silver is less dense than gold, so a cheated crown of the right weight would take up more space — and displace more water.
Whether he really ran through the streets shouting eureka is anyone's guess, but displacement is still how you measure the volume of an awkward shape.
Problem 1
Find the volume of a 5 × 4 × 3 rectangular prism.
Worked answer
60
5 × 4 × 3 = 60
Problem 2
Find the volume of a cube with edge 4 cm.
Worked answer
64
43 = 64 cm3
Problem 3
Find the volume of a cylinder with radius 2 and height 5. Use π ≈ 3.14.
Worked answer
62.8
π r2 = 3.14 × 4 = 12.56
12.56 × 5 = 62.8
Problem 4
Find the surface area of a cube with edge 3.
Worked answer
54
One face: 3 × 3 = 9
6 × 9 = 54
Problem 5
A prism has base area 12 cm2 and height 10 cm. Find its volume.
Worked answer
120
12 × 10 = 120 cm3
Problem 6
A triangular prism has a base triangle with base 6 cm and height 4 cm, and the prism is 10 cm long. Find its volume.
Worked answer
120
B = 12(6)(4) = 12 cm2
V = 12 × 10 = 120 cm3
Problem 7
A cylinder has radius 3 and height 8. Find its volume in terms of π (enter just the number in front of π).
Worked answer
72
9 × 8 = 72
V = 72π
Problem 8
A cylinder has radius 3 and height 8. Find its surface area in terms of π (enter just the number in front of π).
Worked answer
66
Two circles: 2π(9) = 18π
Label: 2π(3)(8) = 48π
18 + 48 = 66, so SA = 66π
Problem 9
Find the surface area of a 5 × 4 × 3 rectangular prism.
Worked answer
94
2(5 × 4) = 40
2(5 × 3) = 30
2(4 × 3) = 24
40 + 30 + 24 = 94
Problem 10
A tank holds 1,680 cm3. Its base is 14 cm by 10 cm. How deep is it?
Worked answer
12
B = 140 cm2
1680 ÷ 140 = 12 cm
Problem 11
A cone and a cylinder have the same base and height. The cone holds what fraction of the cylinder?
- 12
- 13
- 23
- 14
Worked answer
13
Vcone = 13Bh, so it holds one third.
Problem 12
How many 2 cm cubes fit inside an 8 × 6 × 4 cm box?
Worked answer
24
Along each edge: 4, 3, 2
4 × 3 × 2 = 24 cubes
Check: 192 ÷ 8 = 24 ✓
Problem 13
A cube has volume 216 cm3. Find its surface area.
Worked answer
216
63 = 216, so the edge is 6 cm
SA = 6 × 62 = 6 × 36 = 216 cm2
The numbers match here by coincidence — the units do not.
Problem 14
A cylinder has volume 100π and radius 5. Find its height.
Worked answer
4
25π h = 100π
h = 4
Problem 15
A prism's base is a trapezoid with parallel sides 5 and 9 and height 4. The prism is 10 long. Find its volume.
Worked answer
280
B = 5 + 92 × 4 = 28
V = 28 × 10 = 280
Problem 16
A cube's edge is doubled. Its volume is multiplied by what?
Worked answer
8
2 × 2 × 2 = 8
A 3-cube holds 27; a 6-cube holds 216, which is 8 times as much.