# Inner-radius of a (convex) polygon

• NaturePaper
In summary, the conversation discusses whether every convex polygon has an inner-circle and inner-radius. It is mentioned that this is only possible for triangles and regular polygons. There is a question about the formula for the inner-radius in terms of the sides for a convex N-gon with an incircle. The conversation also touches on the usefulness of finding the radius, and it is noted that the radius and center of the circle may differ depending on the side it is touching.
NaturePaper
Hi all,
I want to know whether it is correct that every convex polygon has an inner-circle (& hence an inner-radius). I think it is only possible for triangle and for regular polygon. Am I right?

If there is any convex N-gon having sides a_1,a_2,...,a_N which has an incircle, then what is the formula for the inner-radius in terms of the sides?

Regards,
NaturePaper

If you draw a circle inside the convex polygon, it is not necessary that the circle will touch all the sides.

then..??

So Radius and center of the circle will differ with respect to the side you wish to touch with the circle. then what is the use of finding radius?

@KnowPhysics,
I think I'm missing something.

It is well known that for a triangle with sides a_1, a_2, a_3 there is a (unique) circle inscribed
in the triangle whose radius R is given by some formula involving the sides a_i's.

My question was what about the general situation, i,e for any convex N-gon what will be the situation?

Regards,
NaturePaper

i explained already, it is not necessary that the circle will touch all the sides. So Radius and center of the circle will differ with respect to the side you wish to touch with the circle.

## 1. What is the inner-radius of a convex polygon?

The inner-radius of a convex polygon is a measure of the distance between the center of the polygon and its innermost point. It can also be thought of as the radius of the largest circle that can be inscribed within the polygon.

## 2. How is the inner-radius of a convex polygon calculated?

The inner-radius of a convex polygon can be calculated using the formula:
r = (a * cot(π/n))/2
where r is the inner-radius, a is the length of one side of the polygon, and n is the number of sides in the polygon.

The inner-radius and circumradius are both measures of distance in a polygon, but they are calculated differently. The circumradius is the distance from the center of the polygon to its outermost point, while the inner-radius is the distance to the innermost point. In a regular polygon, the circumradius is always larger than the inner-radius.

## 4. Can the inner-radius of a convex polygon be negative?

No, the inner-radius of a convex polygon cannot be negative. It is a measure of distance and therefore can only have positive values.

## 5. How does the number of sides in a polygon affect the inner-radius?

The number of sides in a polygon directly affects the value of the inner-radius. As the number of sides increases, the inner-radius decreases. This is because a polygon with more sides has a smaller distance between its center and innermost point compared to a polygon with fewer sides.

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