Relation between circumradius and inradius in different triangle
Circumradius and inradius these two terms come from geometry. there is also a unique relation between circumradius and inradius. But relation depends on the condition or types of the polygon. like, if the polygon is square the relation is different than the triangle.
So, we can say that relation between circumradius and inradius will be different for different polygon. for example relation between circumradius and inradius of triangle is entirely different to the relation for equilateral triangle.
Before we discuss circumradius and inradius relation for different polygon, we should know the meaning of these two terms circumradius and inradius in detail. So lets start with Circumradius.
Circumradius is the radius of those circle which is circumscribe (surrounds) the triangle. for example,
consider a triangle ABC. Draw the bisector of all three line segment AB, BC, and AC respectively.
Remember all three bisector passes through the vertices A, B, and C of triangle ABC. Now, take as a center 'O' where all three bisector cuts each other. With center 'O' draw a circle touching all three vertices A, B and C. As shown in figure.
Now, OR = OQ = OB = OP = OA = OC = R (circumradius)
R is known as circumradius. 'O' is known as circumcenter of the circle.
Formula of circumradius is R = abc ∕ 4Δ
Inradius is the radius of those circle which is inscribe (surrounded) by the triangle. For example,
Consider a triangle ABC. Draw the bisector of angle A, B and C respectively. Take point 'O' as center where all three bisectors meet each other. With center 'O' draw a circle touching all three sides of the triangle ABC. As shown in figure.
As we know that,
Inradius is that radius of the circle which is inscribe (surrounded) by the triangle. In other words , those line segment which is perpendicular to the sides of the triangle with center 'O'. Here OR, OQ, and OP are perpendicular to the sides AB, AC, and BC respectively. So, OR = OP = OQ = Inradius.
Now,
OR = OP = OQ = r
'r' is known as inradius of the circle.
'O' is known as incenter of the circle.
Formula of Inradius is, r = (a + b - c) / 2
relation between circumradius and inradius of equilateral triangle
Relation between circumradius and inradius of an equilateral triangle is in such a way that Inradius of a circle is equal to the half of the Circumradius of a circle. As shown in above figure. the formula of inradius for equilateral triangle is same as formula of circumradius. But there is difference of 2 in denominator of inradius's formula. that is given by,
formula of inradius of equilateral triangle = side / 2√3
formula of circumradius of equilateral triangle is side / √3
Inradius = 1/2 ✕ circumradius
∴ circumradius = 2 inradius