Surface Area & Nets — Challenge
Unfold a solid and its surface becomes a flat shape whose area you already know how to find.
What these problems cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Face — A flat surface; a line where two faces meet; a corner point.
- 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.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Pyramid — A base tapering to a point. V = 13 Bh.
- Height — The perpendicular distance from the base to the opposite corner.
- Cubic unit — A cube one unit on every edge.
- Capacity — How much a container holds, often in litres or gallons.
Problem 1
What is the surface area of a cube with side 3?
- 9
- 27
- 36
- 54
- 18
Worked answer
54
A cube has six identical square faces, each of area 3· 3 = 9.
The surface area is 6· 9 = 54 square units.
The value 27 is the volume, which is the side cubed.
Problem 2
What is the surface area of a box measuring 2 by 3 by 4?
- 24
- 48
- 52
- 104
- 26
Worked answer
52
The three different faces have areas 2· 3 = 6, 2· 4 = 8 and 3· 4 = 12.
Each occurs twice, so the surface area is 2(6+8+12) = 2· 26 = 52 square units.
The value 24 is the volume, not the surface area.
Problem 3
How many squares are there in the net of a cube?
- 4
- 5
- 6
- 8
- 12
Worked answer
6
A net is the flat shape that folds up into the solid, so it has one piece for each face.
A cube has six faces, so its net has six squares.
The cube has 8 vertices and 12 edges, which is where those other numbers come from.
Problem 4
What is the surface area of a box measuring 5 by 5 by 2?
- 50
- 70
- 90
- 100
- 130
Worked answer
90
The two square faces have area 5· 5 = 25 each, giving 50.
The four rectangular faces have area 5· 2 = 10 each, giving 40.
The total is 50+40 = 90 square units.
Problem 5
A number cube has opposite face pairs (1,6), (2,5), and (3,4). What is the largest sum of the three faces meeting at a corner?
- 12
- 13
- 14
- 15
- 16
Worked answer
15
A corner contains one face from each opposite pair.
Choose the larger member of each pair: 6+5+4=15.
Problem 6
A cube has a surface area of 96. What is the length of one side?
- 3
- 4
- 6
- 8
- 16
Worked answer
4
The six faces are equal, so each face has area 966 = 16.
A face is a square, so its side is √16 = 4.
Dividing 96 by 6 and stopping gives 16, which is the area of a face rather than its side.
Problem 7
A cube is painted on every face, then cut into 27 identical small cubes. How many of the small cubes have exactly one painted face?
- 6
- 8
- 12
- 1
- 27
Worked answer
6
Cutting into 27 means three small cubes along each edge. A small cube with exactly one painted face must sit in the middle of a face of the big cube, touching no edge. There is one such cube per face, and a cube has six faces — so 6. (The eight corners have three painted faces, and the twelve edge cubes have two.)
Problem 8
Two identical cubes with edge 3 are glued together face to face. What is the surface area of the resulting solid?
- 90
- 108
- 99
- 81
- 54
Worked answer
90
Each cube alone has surface area 6 × 9 = 54, so the two together start at 108. Gluing hides one face on each cube, and each hidden face is 3 × 3 = 9. 108 - 18 = 90. The glued faces are still there; they are simply no longer on the surface.