A cube has 6 faces, 8 corners and 12 edges. Is there a rule that links the three numbers?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Draw a cube on dotted paper
Draw a solid as seen from the front, the side and the top
Check that faces + corners − edges = 2 for a solid
A solid has flat faces. Two faces meet along an edge. Edges meet at a corner. A cube has 6 faces, 12 edges and 8 corners.
Isometric dotted paper has dots in a pattern of triangles. You draw lines that go straight up, up to the right and up to the left. Each step from dot to dot is 1 cm here.
Draw it first
Draw a cube on dotted paper
1 Draw one upright line, 3 steps long.
2 From its foot draw a line up to the right and a line up to the left. Each is 3 steps long. Add an upright line at the end of each.
3 Join the tops to make the top face. Use four lines in all, each 3 steps long.
4 Check: 9 lines, each 3 steps long. The corner at the back is hidden right behind the front top corner.
You can draw a cuboid in the same way. Its three lengths can be different. A cuboid is a box.
A cuboid 4 cm long, 2 cm wide and 3 cm high.
Now count the parts of a cube. Letters name the corners. Dashed lines show the edges you cannot see.
A cube has 8 corners, 12 edges (9 solid and 3 dashed) and 6 faces. This cube is turned a little and is not on dotted paper, so the hidden corner shows.
Take a solid with flat faces and no hole through it. Count its faces, corners and edges. Then work out faces + corners − edges. The answer is always 2.
Worked example
Check that faces + corners − edges = 2 for a cube.
A cube has 6 faces.
It has 8 corners.
It has 9 edges you can see and 3 at the back, so 12 edges.
Faces + corners − edges = 6 + 8 − 12 = 2.
Answer: 2
A square based pyramid: 1 square face and 4 triangle faces.
Worked example
Check that faces + corners − edges = 2 for a square based pyramid.
It has 1 square face and 4 triangle faces. That is 5 faces.
It has 4 corners on the base and 1 at the top. That is 5 corners.
It has 4 edges on the base and 4 edges up to the top. That is 8 edges.
Faces + corners − edges = 5 + 5 − 8 = 2.
Answer: 2
A triangular prism: 2 triangle faces and 3 rectangle faces.
Worked example
Check that faces + corners − edges = 2 for a triangular prism.
It has 2 triangle faces and 3 rectangle faces. That is 5 faces.
It has 3 corners at each end. That is 6 corners.
It has 3 edges at each end and 3 edges between the ends. That is 9 edges.
Faces + corners − edges = 5 + 6 − 9 = 2.
Answer: 2
A solid looks different from the front, from the side and from above. Each of these is a view. You draw a view as a flat shape made of squares.
A stack of 9 cubes on dotted paper.
Worked example
How many squares are in the front view of the stack in the picture?
The front view has one column for each column of the stack. There are 3 columns.
Stand at the front. The left column has a tallest tower of 3 cubes.
The middle column has a tallest tower of 1 cube.
The right column has a tallest tower of 2 cubes.
The front view has 3 + 1 + 2 = 6 squares.
Answer: 6
Each view is a flat shape. The top view shows how many cubes stand on each square.
A net is a flat shape that folds up into a solid. The net of a cube is 6 squares joined along their sides. Not every set of 6 squares folds into a cube.
The net of a cuboid. Dashed lines are folds. Opposite faces have the same shading.
See it three ways
With things
Take a small box, such as a tea box. Count its flat faces, then its edges, then its corners.
As a picture
In symbols
Faces + corners − edges = 6 + 8 − 12 = 2.
Try it
Count the parts of a box
You need: a small box, such as a tea box or a matchbox, a pencil, a sheet of paper.
Count the flat faces of the box. Write the number.
Count the edges. An edge is a line where two faces meet.
Count the corners.
Work out faces + corners − edges.
What you will see: A box has 6 faces, 8 corners and 12 edges. The answer to the sum is 2.
If you cannot try it, this is what happens.
A wrong idea many people have
To count the edges of a solid, count the lines you can see in the picture.
Why it feels right: A picture shows lines, so counting them seems enough.
The test that breaks it: Count the solid lines on the picture of the cube. You find 9. Now count the dashed lines too. There are 3 more, so the cube has 12 edges.
What is true: Edges at the back are drawn as dashed lines. Count the dashed lines too.
Key words
face
a flat surface of a solid.
edge
a line where two faces of a solid meet.
corner
a point where edges of a solid meet.
net
a flat shape that folds up into a solid.
view
the flat shape you see from the front, the side or the top.
Back to your guess
The answer is C: Yes. Faces plus corners take away edges is 2 for every solid like this.
For a cube, 6 + 8 − 12 = 2. The same sum gives 2 for a pyramid and for a prism. It holds for solids with flat faces and no hole through them. Adding the three numbers gives 26 for a cube, but it does not give 26 for other solids.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
A cube has edges of 4 cm. Each step from dot to dot is 1 cm. How many steps long is each line on the dotted paper?
See the answer
4
Each step is 1 cm, so an edge of 4 cm is 4 ÷ 1 = 4 steps long. It passes over 5 dots, but the steps are what count.
If you chose B: You counted the dots. A line that is 4 cm long goes over 5 dots but is only 4 steps long. Go to M3.5.8.
If you chose C: That is the size of the cube drawn in the chapter. This cube has edges of 4 cm. Go to M3.5.8.
How many edges has a square based pyramid?
See the answer
8
The base has 4 edges and 4 more edges go up to the top corner. 4 + 4 = 8.
If you chose A: That is the number of faces, and also the number of corners. Count the lines: 4 on the base and 4 up to the top. Go to M3.5.8.
If you chose C: That is the edges of the base only. Add the 4 edges that go up to the top corner. Go to M3.5.8.
A triangular prism has 5 faces and 6 corners. How many edges has it? Use faces + corners − edges = 2.
If you chose A: You added the 2. The rule says faces + corners − edges = 2, so edges = faces + corners − 2. Go to M3.5.8.
If you chose B: That is 5 + 6. Take away the 2 as well. Go to M3.5.8.
Which of these is a net of a cube?
See the answer
Net C
Fold the four squares in the row round the cube. The square above and the square below fold on to the top and the bottom. Every face is covered once.
If you chose B: A straight row of 6 squares wraps round the cube. The fifth square lands on the first, and the sixth on the second. Go to M3.5.8.
If you chose C: A block of 2 rows of 3 squares will not fold. Two of the squares would land on the same face of the cube. Go to M3.5.8.
Look at the stack of cubes. The white faces on the right are the front. How many squares are in the front view?
See the answer
3
The left column of the front view is as tall as its tallest tower, 2 squares. The right column has 1 square. 2 + 1 = 3 squares.
If you chose A: That counts the cube behind the tower of 2 as a third column, or counts every cube. The front is the white faces on the right. The cube behind adds no square. Go to M3.5.8.
If you chose C: That is the height of the tallest tower. The front view also has a column for the tower of 1 cube. Go to M3.5.8.
A solid has 7 faces and 15 edges. How many corners has it? Use faces + corners − edges = 2.
If you chose A: You added 7 and 15 and took away 2. Put the rule in order: corners = 2 + edges − faces. Go to M3.5.8.
If you chose B: You took 2 away instead of adding it. Corners = 2 + edges − faces. Go to M3.5.8.
A solid has 6 faces and 12 edges. How many corners has it?
See the answer
8
Corners = 2 + edges − faces = 2 + 12 − 6 = 8.
From an earlier chapter
A shape has turning symmetry of order 8. What is the smallest turn that fits it?
See the answer
45°
The 8 turns make one full turn of 360°. Each turn is 360° ÷ 8 = 45°.
Keep it
Copy this into your exercise book.
A solid has faces, edges and corners. Faces + corners − edges = 2. A net folds into a solid. A view is the flat shape you see from the front, side or top.
Draw: A cube on dotted paper, with the 9 edges you can see. Write 6 + 8 − 12 = 2 beside it.
