PENTAGON

, in Geometry, a plane figure consisting of five angles, and consequently sive sides also. If the angles be all equal, it is a regular Pentagon.

It is a remarkable property of the Pentagon, that its side is equal in power to the sides of a hexagon and a decagon inscribed in the same circle; that is, the square of the side of the Pentagon, is equal to both the squares taken together of the sides of the other two sigures; and consequently those three sides will consti-| tute a right-angled triangle. Euclid, book 13, prop. 10.

Pappus has also demonstrated, that 12 regular Pentagons contain more than 20 triangles inscribed in the same circle; lib. 5, prop. 45.

The dodecahedron, which is the fourth regular body or solid, is contained under 12 equal and regular Pentagons.

To find the Area of a Regular Pentagon. Multiply the square of its side by 1.7204774, or by 5/4 of the tangent of 54°, or by . Hence if s de<*> note the side of the Pentagon, its area will be .