Finding a radius from a radius?

Question

Is there a way to convert the inscribed radius to the circumscribed radius of a polygon? And visa versa

The diameters conversion will also work

Answer 1

This I presume is for regular polygon otherwise it is meaningless.

Let the circumscribed radius be R

Let C be the centre and AB be one of sides

join AC and AB
AC = R
now we need to evaluate angle CAB

total angle = (n-2)pi
now each angle = (n-2)pi/n
angle CAB = 1/2(each angle) = (n-2)pi/(2n)

side AC = R
length of perpendicular = side of inscribed radius = r

r/R = sin((n-2)pi/2n)
this is the ratio

Answer 2

First, I assume the polygon is regular with n sides. The inscribed radius is commonly called the "apothem", while the circumscribed radius is just the "radius". I will use "a" to refer to the apothem length and "r" for the radius.

The angle between the radius and the apothem is π/n. The apothem makes a right angle with the side, so the apothem and radius form a right triangle with the side. We can see that

cos(π/n) = a / r

and this is a simple formula in terms of only n relating the two lengths.

Answer 3

The inscribed radius is called the apothem. It is a leg of a right triangle with a (circumscribed) radius as hypotenuse. By finfing the central angle of this triangle (which will be 360 divided by twice the number of sides), and using cosine (adj. leg/hypot.), you can get one from the other

Answer 4

Let the length of each side be x, let the inscribed radius be r, and the circumscribed radius be R.
Then R squared = r squared + (x squared)/4.

Answer 5

This might be useful