Geometry & Figures

Depth · one concept from first principles · step 2 of 4

Face, edge and vertex counts identify a prism

A prism has 10 faces. We need the total of its edges and its vertices.
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How to solve
Strategy Identify Subproblems — Everything about a prism is decided by one number: how many sides its base has. Find that first from the face count, and the edges and vertices follow without any counting in the picture.
1STEP 1

Write the face count in terms of the base

A prism has one side face for each edge of its base, plus the two bases themselves.

(faces) = (base sides) + 2
2STEP 2

Find how many sides the base has

Put 10 in for the faces and take the two bases off.

■ + 2 = 10 → ■ = 8
3STEP 3

Count the edges and the vertices

Each base has 8 edges and one upright joins each pair of matching corners: 8 + 8 + 8 = 24 edges. Corners: 8 per base, 8 + 8 = 16.

(edges) = 8 × 3 = 24, (vertices) = 8 × 2 = 16
4STEP 4

Add the two counts

The question asks for them together.

24 + 16 = 40
Answer
40
Check with Euler's rule for solids: vertices minus edges plus faces should be 2, and 16 - 24 + 10 = 2.
Takeaway

A prism is decided by one number -- how many sides its base has. Find that and the rest follows.

  • Write the face count in terms of the base
  • Find how many sides the base has
  • Count the edges and the vertices
  • Add the two counts
Where next?
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