Math Detours · Grade 5 · Around the corner · 10

Corners, edges and faces

Count them on a cube, a pyramid, a prism. Corners plus faces minus edges is always 2.

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Tonight's idea

A solid with flat sides has three things you can count. A face is a flat side. An edge is a line where two faces meet. A corner is a point where edges meet.

cube8 corners, 12 edges, 6 faces square pyramid5 corners, 8 edges, 5 faces triangle prism6 corners, 9 edges, 5 faces
Dashed edges are round the back. Count them too.

For each solid, add the corners and the faces, then take away the edges.

Cube: \(8 + 6 - 12 = 2\). Pyramid: \(5 + 5 - 8 = 2\). Prism: \(6 + 5 - 9 = 2\).

For any solid with flat faces and no holes through it: corners + faces − edges = 2.

Three very different solids, one answer. This is called Euler's rule, after Leonhard Euler (say "Oiler"), who noticed it more than 250 years ago.

If you know two of the counts, it gives you the third. It also catches mistakes. If your three counts do not give 2, count again.

Watch one: saw off a corner

One corner off, one triangle on.

Saw one corner off a wooden cube.

The old corner goes and the cut makes three new ones: \(8 - 1 + 3 = 10\) corners. The cut is a new triangle face: \(6 + 1 = 7\) faces. The triangle has three new edges: \(12 + 3 = 15\) edges.

Check: \(10 + 7 - 15 = 2\). Corners and faces went up by 3 between them. Edges went up by 3 as well. So the balance did not move.

Your turn

  1. A pyramid with a triangle for its base has four faces, all triangles. Count its corners and its edges. Check the rule.
  2. A prism has two five-sided ends joined by five rectangles. Count its corners, edges and faces. Check the rule.
  3. (a) A solid has 12 corners and 8 faces. How many edges does it have? (b) Sam counts 7 corners, 7 faces and 11 edges on a solid. How do you know Sam has miscounted?
  4. A pyramid has a base with 100 sides. (a) Without drawing it, find its corners, edges and faces. (b) Check the rule. (c) Explain why the rule works for every pyramid, whatever the base.

Stretch: the football

A classic football (soccer ball) is sewn from 12 five-sided patches and 20 six-sided patches. Every edge is a seam shared by two patches. How many edges are there? Now use the rule to find the number of corners.

Answers

  1. 4 corners and 6 edges. \(4 + 4 - 6 = 2\).
  2. 10 corners (5 on each end). 15 edges (5 round each end and 5 joining them). 7 faces (2 ends and 5 rectangles). \(10 + 7 - 15 = 2\).
  3. (a) \(12 + 8 = 20\), and 20 take away the edges must be 2, so there are 18 edges. (b) \(7 + 7 - 11 = 3\), not 2. If the two 7s are right there must be 12 edges, so Sam probably missed one.
  4. (a) 101 corners (100 round the base and the tip). 200 edges (100 round the base and 100 up to the tip). 101 faces (100 triangles and the base). (b) \(101 + 101 - 200 = 2\). (c) Each side of the base brings one corner, one triangle face and two edges. So corners plus faces goes up by 2, and edges go up by 2 as well. The tip and the base are the extra 2.

Stretch. Count the sides of all the patches: \(12 \times 5 + 20 \times 6 = 180\). Each seam was counted twice, once for each patch, so there are \(180 \div 2 = 90\) edges. There are 32 faces. Corners \(+\ 32 - 90 = 2\), so there are 60 corners.

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