117. The Cone's Blueprint · vertex, edge, and two surfaces

Try to break it

Drag A up or down to change the height h, and B left or right to change the base radius r. Try to give the cone more than one vertex, or a second flat face, or two circular edges — impossible. Every cone, tall or short, narrow or wide, has exactly 1 vertex, 1 circular edge, 1 flat base, and 1 curved surface.

How you build it

Construct a cone.

• Draw the base of the cone — a flat ellipse. Click two opposite corners of its bounding box so it looks like a tilted circle.
• Mark the apex (Vertex) directly above the centre of the base — this is the tip of the cone.
• Draw the left slant — a segment from the apex to the leftmost point of the base ellipse.
• Draw the right slant — a segment from the apex to the rightmost point of the base ellipse. The cone is complete.

The proof, step by step

Prove that the solid is a cone — one vertex, one circular base, and one curved surface.

• A cone has exactly one vertex (the apex).
• A cone has exactly one edge (the circular base perimeter).
• A cone has two surfaces: one curved lateral surface and one flat circular base.

Worked example

A solid has one vertex, one edge, and two surfaces. One surface is curved and the other is flat and circular. Identify the solid.

Based on the given properties, the solid is a cone. A cone has a circular base (flat surface), a curved lateral surface, one circular edge, and one vertex.

• Cylinder
• Cone — correct
• Sphere
• Cuboid