Problem
Face, edge and vertex counts identify a prism
A prism has a known number of faces. We need the total of its edges and its vertices.
GeometryExpressions
Your answer
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
The two bases are the only extras.
6.G.A.4Draw A Diagram2STEP 2
Find how many sides the base has
Put 10 in for the faces and take the two bases off.
■ + 2 = 10 → ■ = 8
It is an octagonal prism.
6.EE.B.7Identify SubproblemsA prism with faces has an octagon for its base.
Why?
A prism's faces are its two bases plus one side face for each base edge, and no face is left over, so faces means base edges.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Count the edges and the vertices
Each base contributes its 8 edges, and one upright edge joins each pair of matching corners: 8 + 8 + 8 = 24. The corners are the 8 of each base: 8 + 8 = 16.
(edges) = 8 × 3 = 24, (vertices) = 8 × 2 = 16
24 edges and 16 vertices.
6.G.A.4Look For A Pattern4STEP 4
Add the two counts
The question asks for them together.
24 + 16 = 40
40 in all.
6.G.A.4Identify SubproblemsAnswer
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