How to calculate length of a side of regular polygon in the term of radius of a circle?

Question

the circle is touching all side of the regular polygon

Answer 1

You must specify whether the circle is outside (circumscribed) or inside (inscribed) the polygon!

If the circle is circumscribed:
A triangle can be formed with one edge of the polygon, and two radii of the circle. The angle made by the radii is 360 degrees divided by the number of sides of the polygon. Bisect the triangle (as well as bisecting the interior angle). You can use some trigonometry to determine half the length of the edge of the polygon.
So, where 'p' is the number of sides on the polygon and 'r' is the radius of the circle you'll have:
Length = 2*r*sin( 180/p ).

If the circle is inscribed:
Again you make the same kind of triangle, but this time, the side of the polygon is already bisected. So,
Length = 2*r*sin( 360/p ).

Feel free to email me for any clarifications.

Answer 2

For an inscribed polygon:
Let r be the radius of the circle, n be the number of sides of the regular polygon, and s be the length of a side of the regular n-gon. You can think of the regular n-gon as n triangles, each of which has vertices 2 vertices on a side of the n-gon and the thrid as the center of the circle (you can visualize these as wedges or pieces of pizza).

Now consider one of these triangles. It is isoceles (since the n-gon is regular) and the angle corresponding to the vertex that is the center of the circle is 360/n. That implies that the other two angles sum to 180 - 360/n, and thus are (90 - 180/n) each (note of course that n is a least 3).

So now you can use the Law of Sines, which tells you that:

sin(90 - 180/n)/r = sin(360/n)/s

so s = r*sin(360/n)/sin(90 - 180/n)

Just as a sanity check, lets plug in something we, know like n = 4. If n=4, then the side should be r*sqrt(2). Our formula gives: r*sin90/sin45 = r*sqrt(2). Also note that as n gets smaller, so does s (I like having these kind of checks, since I tend to make a lot of algebra/arithmetic mistakes -- although I do not think I did here =)

For a circumscribed polygon (outside the circle), draw a side of the polygon and the radius that bisects it. Now form a right triangle with half of the side and the bisecting radius. You know the angle opposite ethe half side is (360/n)/2 = 180/n, thus by the definition of tangent (opposite over adjacent), the half side has length = r tan(180/n), so the length of an entire side of the n-gon is:

s = 2rtan(180/n).

Let's try this for an n we know the answer for, if n=4, we should get s = 2r, and plugging n into our formula we get the same thing!

Answer 3

Is the circle on the inside or the outside of the polygon.

If it is on the outside(circumscribed) then the polygon will touch the circle so the distance from the center of the polygon to a corner will be the radius. So make a right triangle out the radius and half of the side and calculate from there.
r=radius
n= number of sides
s=length of side
(n-2)*180/n = one angle of the polygon (A)
From cos(angle) = adjacent side/hypotenuse
cos(A/2)=(s/2)/r
It is A/2 because the angle in the triangle is half of the angle of the polygon, and it is s/2 becuase the side of the triangle is half of the side of the polygon. r is the hypotenuse.
Solve for s

If the circle is on the inside then r becomes the length of the side of the triangle rather than the hypotenuse.

Answer 4

If the circle is the circumcircle, then, the relationship between the radius of the circle and the side of the inscribed regular polygon is given by r = [(s^2/(2(1-cos 360/n))]^1/2, where n is the number of sides, s is the length of one side, and r is the radius.

If the circle is the incircle, which it is since you say the circle touches all sides of the regular polygon, then the relationship between the radius of the incircle and the side of the regular polygon is given by: r = a, where r is the radius of the incircle and a is the apothem of the regular polygon. The apothem is the line segment joining the incenter to the midpoint of any side. If you know the area of the regular polygon, the apothem is equal to twice the area divided by the perimeter of the regular polygon.
The apothem is also = r = s/(2tan(180/n)) where s is the length of one side of the regular polygon and n = the number of sides.

Answer 5

First, since the circles is touching all SIDES of the regular polygon, the circles must inscribe the polygon.

Let a be the length of a side, and ∅ = 360/(2n) = 180/n be the half central angle opposite of one side of polygon.

From a sketch, we can see that (a/2) and r (radius) form a right triangle with
(a/2) / r = tan ∅

Solve for a,
a = 2r tan ∅ = 2r tan(180⁰/n)

Answer 6

Is the circle outside the polygon or inside? It makes a difference!