# Derive the radius of a circle inscribed in an equilateral triangle of side s.

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### 2 Answers

An equilateral triangle with sides equal to s has all sides equal to s.

The radius of the incircle of a triangle with sides a, b and c is given by the formula

r = `sqrt(((S - a)(S - b)(S - c))/S)` where s is the semi-perimeter of the triangle.

In the equilateral triangle a = b = c = s

=> r = `sqrt(((S - s)(S - s)(S - s))/S)`

S - s = `(3*s - 2s)/2 = s/2`

=> r = `sqrt(((s/2)^3/((3*s)/2)))`

=> r = `sqrt((s^3/8)/(3*s/2))`

=> r = `sqrt(s^2/12)`

=> r = `(sqrt 3/6)*s`

**The radius of the inradius of an equilateral triangle with side s is **`(sqrt 3/6)*s`

In an equilateral triangle, the inscribed circle is tangent to all sides and touches the sides at their mid-point. The center of the circle is the intersection of the bisectors of the angle of triangle that are equal to 60 deg.

Side of equilateral triangle = s

Hence the radius = half of side multiplied with tan(bisected angle)

* Radius of Inscribed Circle *= 0.5*s*tan(30)

**= 0.288675*s**