178. The Polyhedron's Faces · Bounded by polygons, not curves

Try to break it

Drag the figure around. Every face stays flat and every edge stays straight — that's what makes it a polyhedron. Try to find a curved face among the polygonal ones; you won't. Polyhedra are built from flat polygonal pieces only.

How you build it

Construct a hexahedron.

• Draw a quadrilateral for the front face of the hexahedron. Click each corner, then double-click the last to close the polygon.
• Draw a second quadrilateral for the back face. Place it slightly behind and to one side of the front face, smaller to show depth.
• Connect the first pair of matching vertices — click one corner of the front face and the corresponding corner on the back face.
• Connect the second pair of matching vertices — click the next front corner and its corresponding back corner.
• Connect the third pair of matching vertices. Three edges down, one to go.
• Connect the fourth and final pair of vertices — close all edges to complete the hexahedron.
• Connect the fifth pair of matching vertices — two edges left.
• Connect the sixth and final pair of vertices — all six connecting edges are now drawn. The hexahedron is complete!

The proof, step by step

Prove that the solid is a polyhedron — every face is a flat polygon.

• A polyhedron is a three-dimensional figure bounded by polygonal regions.
• The shape has flat faces with straight edges and no curved surfaces.
• Therefore, it is a polyhedron.

Worked example

Which of the following solids is a polyhedron?

A polyhedron is bounded by polygonal regions. A cube is bounded by 6 square faces, making it a polyhedron. Cones, cylinders, and spheres have curved surfaces.

• Cone
• Cylinder
• Sphere
• Cube — correct