Stage 8 - Q1 - 2 marks
How many vertices does a cube have?
Hint
Count matching top and bottom vertices.
Stage 8 - Q2 - 2 marks
How many edges does a triangular prism have?
Hint
Separate base edges from joining edges.
Stage 8 - Q3 - 2 marks
A convex polyhedron has 11 faces and 11 vertices. Find its edges.
Hint
Rearrange Euler’s relation.
Stage 8 - Q4 - 2 marks
A convex polyhedron has 8 faces and 12 edges. Find its vertices.
Hint
Make V the unknown in the relation.
Stage 8 - Q5 - 3 marks
Find F, V and E for a pyramid whose base has 11 sides.
Hint
The apex adds one vertex; each base side adds one triangular face.
Stage 8 - Q6 - 2 marks
A prism has 24 vertices. Find its faces and edges.
Hint
Start by finding the number of vertices on one base.
Stage 8 - Q7 - 3 marks
A pupil reports a convex polyhedron with 8 faces, 9 vertices and 16 edges. Show that one count must be wrong.
Hint
Test all three counts together.
Stage 8 - Q8 - 3 marks
A pyramid has 20 edges. Find the base shape, faces and vertices.
Hint
Separate the n base edges and n sloping edges.
Stage 8 - Q9 - 4 marks
A convex polyhedron has equally many faces and vertices and 18 edges. Find both counts and give one possible solid.
Hint
Use the equality to reduce the number of unknowns.
Stage 8 - Q10 - 5 marks
Challenge: can a prism and a pyramid have both the same number of vertices and the same number of edges? Prove your answer for bases with at least three sides.
Hint
Write formulas for both solids before equating their counts.