Volume & Surface Area of 3-D Figures — Challenge
Volume fills the inside — base area times height. Surface area wraps the outside — add up every face.
What these problems 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.
Problem 1
What is the volume of a box measuring 3 by 4 by 5 centimetres, in cubic centimetres?
- 12
- 27
- 47
- 60
- 120
Worked answer
60
The volume of a box is length times width times height.
3· 4· 5 = 60 cubic centimetres.
Problem 2
What is the volume of a cube with side 4 centimetres, in cubic centimetres?
- 16
- 24
- 48
- 64
- 96
Worked answer
64
A cube has all three dimensions equal, so its volume is the side cubed.
43 = 64 cubic centimetres. The surface area would be 96.
Problem 3
A cylinder has radius 3 centimetres and height 7 centimetres. What is its volume in cubic centimetres?
- 21π
- 42π
- 63π
- 147π
- 189π
Worked answer
63π
The volume of a cylinder is the area of its circular base times its height: π r2h.
The base area is π· 32 = 9π square centimetres.
So the volume is 9π· 7 = 63π cubic centimetres.
Problem 4
A cone and a cylinder have the same base and the same height. The cylinder's volume is 90 cubic centimetres. What is the cone's volume in cubic centimetres?
- 10
- 30
- 45
- 60
- 270
Worked answer
30
A cone has one third the volume of the cylinder with the same base and height.
13· 90 = 30 cubic centimetres.
Halving would give 45, which is the ratio for a pyramid against a prism only if the ratio were one half, and it is not.
Problem 5
What is the volume of a box measuring 2 by 2 by 10 centimetres, in cubic centimetres?
- 14
- 20
- 40
- 44
- 100
Worked answer
40
Multiply the three dimensions together.
2· 2· 10 = 40 cubic centimetres.
Problem 6
How many 1-centimetre cubes exactly fill a box measuring 6 by 5 by 4 centimetres?
- 15
- 60
- 120
- 240
- 480
Worked answer
120
Each small cube occupies 1 cubic centimetre, so the count equals the volume in cubic centimetres.
6· 5· 4 = 120 cubes.
Problem 7
What is the volume of a sphere of radius 3 centimetres, in cubic centimetres?
- 9π
- 12π
- 27π
- 36π
- 108π
Worked answer
36π
The volume of a sphere is 43π r3.
With r=3 that is 43π· 27.
43· 27 = 36, so the volume is 36π cubic centimetres.
Problem 8
A cube has a volume of 27 cubic centimetres. What is the length of one side in centimetres?
- 3
- 6
- 9
- 13.5
- 27
Worked answer
3
If the side is s then s3 = 27.
3· 3· 3 = 27, so s = 3 centimetres.
Dividing 27 by 3 gives 9, which is the area of one face, not the side.
Problem 9
A cylinder has a volume of 100π cubic centimetres and a radius of 5 centimetres. What is its height in centimetres?
- 2
- 4
- 5
- 10
- 20
Worked answer
4
The volume is π r2h, so 100π = π· 25· h.
Divide both sides by 25π: h = 4 centimetres.
Problem 10
A tank measures 2 metres by 1.5 metres by 1 metre. How many litres does it hold, given that one cubic metre is 1000 litres?
- 300
- 1500
- 3000
- 4500
- 30000
Worked answer
3000
The volume is 2· 1.5· 1 = 3 cubic metres.
Each cubic metre is 1000 litres, so the tank holds 3· 1000 = 3000 litres.