Polygons are two dimensional shapes with a number of sides. If all the sides of a polygon, the polygon is said to be regular polygon.

Next to triangles and quadrilaterals, the prominent polygons are pentagons and hexagons. A Bucky ball is a combination of 12 pentagons and 20 hexagons. As the names suggest, a pentagon has five sides and a hexagon has six sides.

The regular pentagons and regular hexagons have some important and special properties and let us take a look at them.

## Properties of Pentagons

Take a look at the following regular pentagon.

O is the center of the pentagon, *s* is the measure of the side and *a* is the measure of the apothem.

From the formula for the sum of interior angles, the sum of the interior angles of a pentagon is (180^{0})(5 – 2) = 540^{0} and for a regular pentagon, each interior angle is, (540^{0})/(5) = 108^{0}

The center of the pentagon is the intersection of all the perpendicular bisectors of the sides and it is also the center of the circumscribing circle of the pentagon. Therefore, the line joining the center to a vertex is the radius R of the circle.

The radius R also bisects the interior angle.

The apothem of the pentagon can be worked out as,

a = (s/2)(tan 54^{0} )

The area of the regular pentagon can be worked out from the formula,

A = (1/2)(P)a = (1/2)(5s)( (s/2)(tan 54^{0}) = (5/4)(s^{2})(tan 54^{0})

## Properties of Heagons

Now let us consider the case of a regular hexagon from geometry help

O is the center of the hexagon, *s* is the measure of the side and *a* is the measure of the apothem.

From the formula for the sum of interior angles, the sum of the interior angles of a pentagon is (180^{0})(6 – 2) = 720^{0} and for a regular pentagon, each interior angle is, (720^{0})/(6) = 120^{0}

The center of the hexagon is the intersection of all the long diagonals and it is also the center of the circumscribing circle of the pentagon. Therefore, the line joining the center to a vertex is the radius R of the circle.

The radius R bisects the interior angle and the apothem bisects the side.

The apothem of the hexagon can be worked out as,

a = (s/2)(tan 60^{0} ) = ?3(s/2) = (?3/2)(s)

The area of the regular hexagon can be worked out from the formula,

A = (1/2)(P)a = (1/2)(6s)(?3/2)(s) = (3?3/2)(s^{2})

## Properties of Polygons – Construction of Polygons

The properties of the polygons are used to construct the polygons. The exterior angle properties is one of them. Let us see how it is used to construct a pentagon. The method is exactly the same for a hexagon also.

For a regular pentagon, the exterior angle at any vertex is (180^{0} – 108^{0}) = 72^{0}

Draw a horizontal line *l* and mark a point A. With A as center draw an arc of radius (equal to the measure of the side of the pentagon) to cut the line at B. Mark the point B.

At point B draw a ray *m* at an angle of to the *left* of the line *l*. With B as center strike an arc of the same radius to cut the ray *m* at C. Mark the point C.

Repeat the same steps to mark the points D and E.

Finally join E and A.

ABCDE is the required pentagon.