Problem
Face, edge and vertex counts identify a pyramid
For a pyramid, the number of faces plus the number of edges is known. We need how many vertices it has.
GeometryExpressions
Your answer
How to solve
Strategy Identify Subproblems — Write both counts in terms of the same unknown -- the number of sides the base has -- so that the given sum becomes one equation in one letter.
1STEP 1
Write the faces and edges from the base
A pyramid has one side face per base edge plus the base itself, and each base edge also has a slanted edge running up to the apex.
(faces) = ■ + 1, (edges) = ■ × 2
Both counts come from the same number.
6.G.A.4Draw A DiagramA pyramid on an -sided base has faces and edges.
Why?
Each base edge carries exactly one triangular side face, and the base itself is one face more.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Each base corner sends exactly one slanted edge up to the apex, so the edges are the base edges plus slanted ones.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Turn the given sum into one equation
Add the two expressions and set the total to 28.
(■ + 1) + ■ × 2 = ■ × 3 + 1 = 28
One letter, one equation.
6.G.A.4Convert To Algebra3STEP 3
Solve for the base
Take the 1 off both sides and divide by 3.
■ × 3 = 27 → ■ = 9
It is a nonagonal pyramid.
6.EE.B.7Identify Subproblems4STEP 4
Count the vertices
The base has 9 corners and the apex is one more.
9 + 1 = 10
10 vertices.
6.G.A.4Identify SubproblemsAnswer
10
Put it back: a nonagonal pyramid has 10 faces and 18 edges, and 10 + 18 = 28 as given.
Takeaway
Write every count in terms of the same one unknown, and a puzzle turns into an equation.
- Write the faces and edges from the base
- Turn the given sum into one equation
- Solve for the base
- Count the vertices