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