Verify solid shapes using Euler's formula, and understand how 2D nets fold into 3D solids like prisms and pyramids.
Euler's formula, F + V − E = 2, holds true for any solid with flat faces. It's a quick way to check whether a described solid is actually possible, or to find a missing count.
A solid has 6 faces and 12 edges. How many vertices should it have?
F + V − E = 2 → 6 + V − 12 = 2 → V = 8 (this matches a cube)
Could a solid have 5 faces, 8 edges, and 6 vertices?
Check: 5 + 6 − 8 = 3, not 2 — so this solid is not possible
A prism has two identical, parallel polygon bases connected by rectangles. A pyramid has one polygon base and triangular faces meeting at a single point (the apex). Their nets show all faces unfolded flat.
How many rectangular faces does a triangular prism's net have?
3 rectangles, connecting the two triangular bases
How many triangular faces does a square pyramid's net have?
4 triangles, one for each side of the square base
1. A solid has 8 faces and 12 vertices. Use Euler's formula to find its edges.
2. Could a solid have 4 faces, 4 vertices, and 6 edges?
3. How many total faces does a triangular prism have (bases plus rectangles)?
4. How many total faces does a pentagonal pyramid have?
5. A hexagonal prism has 2 hexagonal bases. How many rectangular side faces does it have?