Regular tetrahedron, octahedron, dodecahedron

The idea

A regular tetrahedron has all its faces equilateral triangles, an octahedron is made by joining two square pyramids at their square bases, and a dodecahedron is a solid whose faces are all pentagons.

Faces that are all alike

A cube has 6 faces, and every face is a square. A regular tetrahedron has 4 faces, and every face is an equilateral triangle.

Is there a solid whose faces are all pentagons?

• No. Pentagons cannot close up into a solid
• Yes, there is such a solid
• Only if some of the faces are squares

Mark every face

Choose a solid and tap its faces to mark them. Drag the solid to turn it: some faces are on the back. The octahedron starts as two pieces. Slide Join to put them together.

Three solids with faces all alike

A regular tetrahedron has all its faces equilateral triangles. An octahedron is made by joining two square pyramids at their square bases. A dodecahedron is a solid whose faces are all pentagons.

The name counts the faces: tetra is 4, octa is 8 and dodeca is 12. The two square bases of the pyramids are stuck together inside the octahedron, so only their 4 + 4 sloping triangles are faces.

• Solid | Faces | Every face is
• Regular tetrahedron | 4 | equilateral triangle
• Cube | 6 | square
• Octahedron | 8 | equilateral triangle
• Dodecahedron | 12 | pentagon

Notes

A regular tetrahedron has 4 faces, all equilateral triangles. An octahedron is two square pyramids joined at their square bases, with 8 triangular faces. A dodecahedron has 12 faces, all pentagons.

Check yourself

Which of these solids has faces that are all pentagons?

Two square pyramids are joined at their square bases to make an octahedron. How many triangular faces does the octahedron have?

Answer: 8 faces

4 triangles + 4 triangles = 8 faces. The two square bases are joined together inside, so the octahedron has no square face.

A dodecahedron has 12 faces. Each face is a pentagon with 5 edges, and every edge is shared by two faces. How many edges does it have?

Answer: 30 edges

12 × 5 = 60 edge-sides, and every edge is counted twice, so 60 ÷ 2 = 30 edges.

Ravi looks at this solid. It has 4 faces and all of them are triangles, so he says, "This is a regular tetrahedron." Is he right?

• Solid A. A is an octahedron. All 8 of its faces are triangles.
• Solid B — correct. Yes! B is a dodecahedron: all 12 of its faces are pentagons.
• Solid C. C is a cube. All 6 of its faces are squares.
• Yes. Any solid with four triangular faces is a regular tetrahedron.. Four triangles make a tetrahedron, but "regular" asks for more. Look at the sides: these triangles are not all equilateral.
• No. It leans, so it is not a tetrahedron at all.. It still has 4 triangular faces: it is a triangular pyramid, which is a tetrahedron. Leaning does not change that.
• No. It is a tetrahedron, but a regular one needs all its faces to be equilateral triangles. — correct. Yes! A regular tetrahedron has all four faces equilateral. Here the triangles have different sides.