Faces, vertices and edges of solids
Build seven solids from straws and clay: a cube, a cuboid, prisms, pyramids, a tetrahedron and an octahedron. Count their faces, vertices and edges, and find the number that F + V − E always gives.
Core concept
A solid has faces (flat surfaces), edges (where faces meet) and vertices (corners); for every solid with flat faces like these, Euler's formula says F + V − E = 2
Topic: Solid objects
A solid object takes up space. Its flat surfaces are faces, the lines where two faces meet are edges, and the corners where edges meet are vertices. A cube has 6 square faces, 12 edges and 8 vertices. Prisms have the same shape at both ends, joined by rectangles; pyramids have a base and triangles meeting at one top vertex. The regular solids, such as the tetrahedron (4 triangles) and the octahedron (8 triangles), have identical regular faces. Leonhard Euler noticed something surprising about all of them.
What you will learn
- Identify the faces, vertices and edges of a solid.
- Name cubes, cuboids, prisms, pyramids, tetrahedra and octahedra.
- Verify Euler's formula F + V − E = 2.
Before you start
- Know the names of plane shapes.
Materials
- Straws or toothpicks and clay or chickpeas soaked overnight
Equipment
- A box, a dice, a pyramid-shaped tea bag or other solid objects
Procedure
- Record the cube.
- Record four or more other solids.
- Record all seven solids.
How this simulation is simplified
- The solids have flat faces and no holes through them.