A pentagon is a geometrical shape. It is any polygon with five sides. The internal angles in a pentagon total 540 degrees.

There are two commonly used units of measurement for angles. The more familiar unit of measurement is a "degree". A circle is divided into 360 equal degrees, so that a right angle is 90°.

The second common measurement for angles is the radian. A radian is defined by the relation s=r*angle. In otherwords, the arclength of a circle is defined by its radius. This is very useful. So, for the unit circle with radius = 1 inch, a point on the edge of the circle travels also 1 inch. Then the angle swept out by the point traveling on the circle is 1 radian.

A pentagon is a plane geometrical figure bounded by 5 straight lines . It has, therefore, five sides, five vertices , five angles and 5 diagonals.

Measure of angle:

Angle is a measure of mutual inclination of two rays.Let OA and OB be two rays. Then the angle between the rays is angle AOB. If you take the compass with centre as O and with radius r cut both OA and OB at A' and B', then OA' = OB' =r.

Then angle AOB = (arc lenth A'B' of the circle with radius r)/r is the measure of the angle in radians. You can convert this into degrees by multiplying by **180/pi**, where pi is the constant fraction = circumference/diameter of a circle and is approximately 3.1416 . An angle can measure from 0 to 2pi radians or from o degrees to 360 degrees. When two rays meet actually there are two angles. But we normally take lesser of the two. The anticlock wise measurement is taken positive measure of angle and the clockwise negative in direction oriented situations.

Measure of pentagon angles:

Take any point O inside the boundary of the pentagon ABCDE with vertices A,B,C,D and E. Join OA, OB,OC, OD and AE. Now ABO is a triangle. Like that each of the sides makes 5 triangles with O. Therefore ,the sum of all the angles of 5 triangles = 5*180 = 900 degrees. If you subtract 360 degree arround O you get the sum of the 5 angles at the vertices of the petagon = 900-360 = 540 degrees or 6 right angles.

So, the sum of the angles of the pentagon at the vertices=angles A+B+C+D+E = 540 degree= 6 right angles (by this proof).

If all the 5 sides of a pentagon are equal and all the angles are equal it is called a regular pentagon, otherwise it is an irregular pentgon.

In a regular pentagon the angle at each vertices = 540/5 =108 degrees.