geometry · Year 6 · Suggested learning sequence
Euler's formula
For any solid with flat faces and straight edges (a polyhedron), the faces (F), vertices (V) and edges (E) are linked: F + V − E = 2. Check it on a cube: 6 + 8 − 12 = 2. So if you know two of the counts, you can find the third. To find edges, add the faces and vertices, then take 2. To find faces, take the vertices from the edges, then add 2. To find vertices, take the faces from the edges, then add 2. Always check by putting all three numbers back into the formula: the answer must be exactly 2.
Sample questions
A pentagonal pyramid has 6 faces and 6 vertices. Use Euler's formula F + V − E = 2 to find how many edges it has.
- F + V − E = 2, so E = F + V − 2.
- E = 6 + 6 − 2 = 10.
- It has 10 edges.
Answer: 10
A square pyramid has 5 faces and 5 vertices. Use Euler's formula F + V − E = 2 to find how many edges it has.
- F + V − E = 2, so E = F + V − 2.
- E = 5 + 5 − 2 = 8.
- It has 8 edges.
Answer: 8
A pentagonal prism has 7 faces and 10 vertices. Use Euler's formula F + V − E = 2 to find how many edges it has.
- F + V − E = 2, so E = F + V − 2.
- E = 7 + 10 − 2 = 15.
- It has 15 edges.
Answer: 15