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Faces, edges and corners (vertices) of solids

Geometry · level 2, lesson 3 of 6

A question to start

Pick up a dice (or any cube). Count its flat sides. Count its corners. Count its edges, the straight lines where two flat sides meet.

You should get 6 flat sides, 8 corners and 12 edges. Now do a strange sum: add the flat sides and the corners. 6 + 8 = 14. That is exactly 2 more than the number of edges, 12.

Is that a coincidence? Try a pyramid, a tent shape, a box. The same thing happens every time! In this lesson you will learn to count the parts of solid shapes carefully, and you will meet one of the most famous patterns in all of geometry.

The idea

Three parts of a solid

In level 1 you met solid shapes: shapes that take up space, like cubes, boxes, balls and cones. Solids have three kinds of parts.

1 2 3 4 5 6 7 8 8 corners (vertices) a face an edge a face and an edge
  • A face is a flat side of a solid. A cube has 6 faces, and each one is a square. (The green square on the right is one face.)
  • An edge is a line where two faces meet. Run your finger along it: it is the sharp, straight part. (The thick line on the right is one edge.)
  • A vertex is a corner, a point where edges meet. More than one are called vertices. In everyday words, we just say corners. (The numbered dots on the left are the vertices.)

Notice that these are the same words you used for flat shapes, but one size bigger. A flat shape has sides that are lines and corners that are points. A solid has faces that are flat shapes, edges that are lines, and corners that are points.

Counting a cube carefully

When you count the parts of a solid, the hardest thing is that you cannot see the back. Some faces, edges and corners are hidden. In the picture on the left, the hidden edges are drawn with dashes, and hidden corner 8 is at the back. Good counting needs a plan.

Faces. Count top and bottom (2), front and back (2), left and right (2). That is 6 faces.

Corners. Count the top layer of corners (4, numbered 1, 2, 5 and 6), then the bottom layer (4, numbered 3, 4, 7 and 8). That is 8 corners.

Edges. Count the edges around the top face (4), the edges around the bottom face (4), and the edges going up and down between them (4). That is 12 edges.

Here is a second way to count the edges, to check. Each of the 6 faces is a square, with 4 edges. That gives 6 × 4 = 24. But every edge is where two faces meet, so every edge was counted twice, once for each face. So the real number is 24 ÷ 2 = 12. The two ways agree, so we can trust the answer.

And the same trick for corners: 6 faces, each with 4 corners, gives 24. But at each corner of a cube, three faces meet (look at any corner of a dice: you can see three faces touching it). So every corner was counted 3 times: 24 ÷ 3 = 8. It agrees again!

More solids

box square pyramid triangle prism cylinder cone sphere

A prism is a solid with two matching ends that are the same polygon, joined by rectangles. A box is a prism with rectangle ends. A triangle prism has triangle ends, like a tent or a chocolate bar box shaped like a long triangle.

A pyramid has one polygon at the bottom (the base) and triangles that lean in to meet at one point at the top. A square pyramid has a square base.

Here are the counts. Count them yourself on the pictures, or better, on real objects.

Solid Faces Corners Edges
cube 6 (squares) 8 12
box 6 (rectangles) 8 12
square pyramid 5 (1 square, 4 triangles) 5 8
triangle prism 5 (2 triangles, 3 rectangles) 6 9
triangle pyramid 4 (all triangles) 4 6

Let us check the square pyramid by counting. Faces: the square base, plus one triangle on each of its 4 sides, 1 + 4 = 5. Corners: 4 around the base, plus the point at the top, 4 + 1 = 5. Edges: 4 around the base, plus 4 going up to the point, 4 + 4 = 8.

Solids with curved parts

Cylinders, cones and spheres have curved surfaces. A curved surface is not called a face by most mathematicians, because a face must be flat. So we describe them like this:

Solid Flat faces Curved surfaces Edges Corners
cylinder 2 (circles) 1 2 (curved, round the circles) 0
cone 1 (a circle) 1 1 (curved) 0, but it has 1 point at the top (the apex)
sphere 0 1 0 0

The wheels of a car, a bicycle or a Mars rover are short cylinders: the rolling part is the curved surface, and the flat faces are on each side.

The famous pattern

Now look again at the solids with flat faces only. Add the faces and the corners, and compare with the edges.

Solid Faces + Corners Edges Difference
cube 6 + 8 = 14 12 2
square pyramid 5 + 5 = 10 8 2
triangle prism 5 + 6 = 11 9 2
triangle pyramid 4 + 4 = 8 6 2

Every time, faces + corners is 2 more than edges. This is called Euler's rule, after Leonhard Euler (say "OY-ler"), a great mathematician who lived about 300 years ago and wrote about it. It works for every solid made of flat faces with no holes through it.

Can we see why, at least for prisms? Think of a prism whose ends have some number of sides. Let us try 5 (a pentagon prism, like a tall box with pentagon ends).

  • Faces: 2 ends, plus one rectangle for each side of the pentagon: 2 + 5 = 7.
  • Corners: 5 on the top pentagon and 5 on the bottom: 5 + 5 = 10.
  • Edges: 5 round the top, 5 round the bottom, 5 going up and down: 5 + 5 + 5 = 15.

Faces + corners = 7 + 10 = 17 = 15 + 2. It works! And look at the reason. The edges are "three fives". The corners are "two fives", and the faces are "one five and 2 more". So faces + corners is three fives and 2 more, which is the edges and 2 more. If you change 5 to any other number, the same thing happens. That is the start of a real proof.

Worked examples

Example 1. Count a triangle prism. A tent has a triangle at the front and back. How many faces, corners and edges?

  1. Faces: 2 triangle ends, plus one rectangle for each side of the triangle (3). Total 2 + 3 = 5.
  2. Corners: 3 on the front triangle, 3 on the back. Total 6.
  3. Edges: 3 round the front, 3 round the back, 3 joining front to back. Total 9.
  4. Check with Euler's rule: 5 + 6 = 11, and 11 is 9 + 2. Yes.

Example 2. Which solid? I have 5 faces. One is a square and 4 are triangles. What am I?

  1. A square at the bottom and triangles on every side, meeting at the top.
  2. That is a square pyramid, like the pyramids of Egypt.

Example 3. Use the rule to find a missing count. A solid has 8 faces and 6 corners. How many edges must it have?

  1. Euler's rule: faces + corners = edges + 2.
  2. 8 + 6 = 14.
  3. So edges + 2 = 14, which means edges = 12.
  4. The solid has 12 edges. (This solid really exists: it is called an octahedron, and it looks like two square pyramids stuck together base to base. Check: 4 + 4 triangles = 8 faces; 4 corners around the middle + 1 top + 1 bottom = 6 corners; 4 edges round the middle + 4 up + 4 down = 12 edges.)

Try it

  1. How many faces does a box (like a cereal box) have? What shape are they?
  2. How many edges does a square pyramid have?
  3. How many corners does a triangle prism have?
  4. Which solid has 2 flat circle faces and 1 curved surface?
  5. A triangle pyramid has 4 faces, all triangles. How many corners and edges? Check with Euler's rule.
  6. A dice is a cube with dots on it. How many faces does it have? How many edges?
  7. A hexagon prism (like an unsharpened pencil) has hexagon ends. How many faces, corners and edges?
  8. A solid has 6 corners and 9 edges. How many faces does it have? Which solid could it be?
Answers
  1. 6 faces, all rectangles (top, bottom, front, back, left, right). Some boxes have two square ends, but the count is still 6.
  2. 8: 4 around the square base and 4 going up to the point.
  3. 6: 3 on the front triangle and 3 on the back triangle.
  4. A cylinder, like a can.
  5. Corners: 3 round the base triangle plus 1 at the top = 4. Edges: 3 round the base plus 3 going up to the top = 6. Euler: 4 + 4 = 8, and 8 = 6 + 2. It works.
  6. A dice is a cube, so it has 6 faces and 12 edges. The dots do not change the shape!
  7. Faces: 2 hexagon ends + 6 rectangles = 8. Corners: 6 on top + 6 on the bottom = 12. Edges: 6 round the top + 6 round the bottom + 6 up and down = 18. Check: 8 + 12 = 20 = 18 + 2.
  8. Faces + 6 = 9 + 2 = 11, so faces = 11 − 6 = 5. A solid with 5 faces, 6 corners and 9 edges is a triangle prism (like a tent).

Common mistake

Forgetting the hidden parts. In a picture of a cube, you can only see 3 faces, 7 corners and 9 edges. The others are round the back. Always count with a plan (top layer, bottom layer, in between), or hold the real object and turn it.

Counting the same edge twice. When you count edges face by face, each edge belongs to two faces. If you do not halve your total, you get 24 for a cube instead of 12.

Where it leads