Lesson Inscribed and circumscribed polygons

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In this lesson we are going to discuss the concept of Incircle and Circumcircle of a polygon and its properties.

Incircle of a <A HREF=http://www.algebra.com/algebra/homework/Polygons/Types-of-Polygon-and-properties.lesson>Polygon</A> :

It is the largest circle that will fit inside a polygon that touches every side.

The incircle of a regular polygon is the largest circle that will fit inside the polygon and touch each side in just one place, hence each of the sides is a tangent to the incircle. If the number of sides is 3, then the result is an equilateral <A HREF=Triangle.wikipedia>triangle</A> and a circle inscribed in it.

The formula for calculating incircle of a polygon are:

If the length of a side is given:

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the inradius is given by the formula:

{{{Inradius = s/(2*tan(pi/n))}}}

where,

s is the length of any side
n is the number of sides
{{{pi}}} is <A HREF=Pi.wikipedia>PI</A>, approximately 3.142
<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/solving-trigonometric-equations.lesson>TAN</A> is the tangent function calculated in <A HREF=Radian.wikipedia>radians</A>

If the radius or circumradius is given:

If you know the radius (distance from the center to a vertex):

{{{Inradius=r*cos(pi/n)}}}

where,

r is the radius (circumradius)
n is the number of sides
{{{pi}}} is PI, approximately 3.142
<A HREF=Cosine.wikipedia>COS</A> is the cosine function calculated in radians

Irregular polygons it is difficult to have an incircle or even a center. If you were to draw a polygon at random, it is unlikely that there is a circle that has every side as a tangent. An exception is a 3-sided polygon (triangle). All triangles always have an incircle.

Circumcircle of a Polygon

It is the circle that passes through each vertex of the regular polygon.The circumcircle of a regular polygon is the circle that passes through every vertex of the polygon.

The radius of the circumcircle (the circumradius) of a regular polygon is exactly the same as the radius of the polygon. That is, the distance from the center to any vertex. The formulas below are the same as for the radius.

If the length of a side is given:

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the radius is given by the formula:

{{{radius= s/(2*sin(pi/n))}}}

where,

s is the length of any side
n is the number of sides
{{{pi}}} is PI, approximately 3.142
<A HREF=Sine.wikipedia>SIN</A> is the sine function calculated in radians

If the inradius is given:

If you know the inradius (distance from the center to the midpoint of a side):

{{{radius=a/cos(pi/n)}}}

where,

a is the apothem (inradius)
n  is the number of sides
{{{pi}}} is PI, approximately 3.142
COS  is the cosine function calculated in radians

Irregular polygons are not usually considered as having a circumcircle. If you draw a polygon at random, it is unlikely there will be a circle that passes through every vertex.

However, the reverse can happen. That is, you can start with a circle and then inscribe a polygon inside it.

For further reading refer to <A HREF=Circumscribed-circle.wikipedia>wikipedia</A> .