Pick one before you read on. It is fine to be wrong.
By the end you can
Find the surface area of a cuboid from its net
Find how much paper covers a box with no gaps
You want to wrap a box in paper with no gaps and no overlaps. The paper must cover the whole outside. The total area of all the faces of a solid is its surface area.
A cuboid has six faces, and every face is a rectangle. They come in three pairs of equal faces. The top and bottom are a pair. The front and back are a pair. The left and right are a pair.
Cut a box along some edges and lay it flat. The flat shape is a net. Its six rectangles are the six faces. Find the area of each rectangle, then add them.
Draw it first
Add the six faces of a net
1 Draw the net of the box. It has six rectangles. Matching faces are shaded the same way.
2 Write the lengths on it: 5 cm, 3 cm and 2 cm.
3 Find the area of each face and add them. The total is 62 cm².
Faces come in pairs, so you only need three areas. Work out one face from each pair. Add the three areas. Then double the total.
Worked example
A box is 5 cm long, 3 cm wide and 2 cm high. What is its surface area?
The top and bottom are each 5 cm by 3 cm: 5 × 3 = 15 cm².
The front and back are each 5 cm by 2 cm: 5 × 2 = 10 cm².
The left and right are each 3 cm by 2 cm: 3 × 2 = 6 cm².
Add one of each pair: 15 + 10 + 6 = 31 cm². Then double it: 2 × 31 = 62 cm².
Answer: 62 cm²
Each rectangle appears twice on the box.
A cube is quicker, because all six faces are the same square.
Worked example
A cube has edges of 4 cm. What is its surface area?
Every face is a square, 4 cm by 4 cm.
One face has an area of 4 × 4 = 16 cm².
A cube has 6 faces: 6 × 16 = 96 cm².
Answer: 96 cm²
Six squares of 16 cm² each make 96 cm².
Surface area answers real questions. Paper that covers a box with no gaps is exactly its surface area. A box with no lid has one face fewer, so you take that face away.
Worked example
A gift box is 10 cm long, 6 cm wide and 5 cm high. How much paper covers it with no gaps?
The top and bottom are 10 × 6 = 60 cm² each.
The front and back are 10 × 5 = 50 cm² each.
The left and right are 6 × 5 = 30 cm² each.
One of each pair adds to 60 + 50 + 30 = 140 cm². Double it: 2 × 140 = 280 cm².
Answer: 280 cm²
The paper for the gift box is 280 cm². Not to scale.
See it three ways
With things
Wrap a small box in paper. Cut the paper along the edges so that it lies flat. You get six rectangles.
You need: a small box, such as a matchbox or a tea box, a ruler, paper, scissors, a pencil.
Measure the length, the width and the height of the box in cm.
Work out the area of one face from each pair. Add them and double the total.
Cover the box with paper, with no gaps and no overlaps. Cut the paper into the six faces.
Lay out the six pieces. Check that matching faces are the same size.
What you will see: The six pieces make three matching pairs. Their areas add up to the total you worked out. A box 5 cm by 3 cm by 2 cm needs 62 cm² of paper.
If you cannot try it, this is what happens.
A wrong idea many people have
Surface area and volume are the same thing.
Why it feels right: Both are about a box, and both come from its length, width and height.
The test that breaks it: A box 4 cm by 3 cm by 2 cm holds 24 cm³. Its surface area is 52 cm². The numbers are different, and so are the jobs: one counts cubes inside, the other counts squares outside.
What is true: Volume is the space inside, in cm³. Surface area is the area of the outside, in cm².
Key words
surface area
the total area of all the faces of a solid shape.
faces
the flat sides of a solid shape.
net
a solid shape cut along some edges and laid flat.
Back to your guess
The answer is B: The area of all six faces of the box added together.
Wrapping paper must cover every face of the box. So you add the areas of all six faces. The volume counts the space inside, which is a different job.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
What is the surface area of this box?
See the answer
52 cm²
One of each pair: 4 × 3 = 12 cm². 4 × 2 = 8 cm². 3 × 2 = 6 cm². They add to 26 cm². Double it: 2 × 26 = 52 cm².
If you chose B: You multiplied 4 × 3 × 2. That is the volume, the space inside, and it is in cm³. Surface area adds the areas of the faces. Go to M3.6.7.
If you chose C: You added one face from each pair: 12 + 8 + 6. Every face appears twice on the box, so double it. Go to M3.6.8.
A cube has edges of 3 cm. What is its surface area?
See the answer
54 cm²
One face is 3 × 3 = 9 cm². A cube has six equal faces: 6 × 9 = 54 cm².
If you chose A: That is one face. A cube has six faces. Go to M3.6.8.
If you chose B: You multiplied 3 × 3 × 3. That is the volume, in cm³. Surface area adds the areas of the six faces. Go to M3.6.7.
A cuboid has three different lengths. How many pairs of equal faces does it have?
See the answer
3 pairs
A cuboid has six faces. Opposite faces match, so they make 6 ÷ 2 = 3 pairs.
If you chose A: Count again. The top and bottom match, the front and back match, and the left and right match. That is more than two. Go to M3.6.8.
If you chose C: There are six faces in all. Two faces make a pair, so there are fewer pairs than faces. Go to M3.6.8.
A box is 8 cm long, 5 cm wide and 3 cm high. How much paper covers it with no gaps?
See the answer
158 cm²
One of each pair: 8 × 5 = 40 cm². 8 × 3 = 24 cm². 5 × 3 = 15 cm². They add to 79 cm². Double it: 2 × 79 = 158 cm².
If you chose B: You added one face from each pair: 40 + 24 + 15. Every face appears twice, so double it. Go to M3.6.8.
If you chose C: You multiplied 8 × 5 × 3. That is the volume of the box, not the paper. Go to M3.6.7.
A box has no lid. It is 6 cm long, 4 cm wide and 3 cm high. How much card makes it?
See the answer
84 cm²
With a lid the box needs 2 × (24 + 18 + 12) = 108 cm². The lid is 6 × 4 = 24 cm². Without it: 108 − 24 = 84 cm².
If you chose A: That covers a box with a lid. This box has no lid, so take the top face away. Go to M3.6.8.
If you chose B: You multiplied 6 × 4 × 3. That is the volume of the box. Card for the faces is an area. Go to M3.6.7.
Two boxes hold 64 cm³ each. One is 4 by 4 by 4. The other is 8 by 4 by 2. Which needs more paper to cover it?
See the answer
The 8 by 4 by 2 box
The 4 by 4 by 4 box has 2 × (16 + 16 + 16) = 96 cm². The 8 by 4 by 2 box has 2 × (32 + 16 + 8) = 112 cm². So the flatter box needs more paper.
If you chose A: The cube has 96 cm² of faces. The long flat box has 112 cm². The same volume can have a different surface area. Go to M3.6.8.
If you chose C: The same volume does not mean the same surface area. Work out the six faces of each box and compare. Go to M3.6.8.
A box is 7 cm long, 4 cm wide and 3 cm high. What is its surface area?
See the answer
122 cm²
One of each pair: 7 × 4 = 28 cm². 7 × 3 = 21 cm². 4 × 3 = 12 cm². They add to 61 cm². Double it: 2 × 61 = 122 cm².
From an earlier chapter
A box is 5 cm long, 4 cm wide and 3 cm high. What is its volume?
See the answer
60 cm³
The volume is length × width × height: 5 × 4 × 3 = 60 cm³.
Keep it
Copy this into your exercise book.
A cuboid has 6 faces in 3 pairs. Surface area = 2 × (l × w + l × h + w × h). It is the paper that covers the box with no gaps, in cm². Volume is a different measure, in cm³.
Draw: The net of a box with its six rectangles numbered, and the three pairs shaded alike.
