Question

what is the formula for finding radius of a circle inscribed in an equilateral triangle???

Answer 1

Answer:

HI MATE !

An equilateral triangle ABC, AB= BC = AC = a unit, AM is an altitude to BC from A also bisecting BC. Since triangle ix equilateral so AM will be a median, & an angle bisector too, with O its centroid also its incentre. So inscribed circle is with centre O & radius = OM = r

In right tri AMB

.

AB² = BM² + AM² ( by Pythgoras law)

Or, a² = a²/4 + AM²

=> AM² = a² - a²/4 = 3a²/4

=> AM = (√3a) /2 unit

Now, we divide this median length into 3 equal parts. & radius ( OM) will have its one part.

Since AM is median . So centroid O divides each median in the ratio 2:1

AO : OM = 2:1

=> AO = 2x, OM = x

=> AM = 3x

.

=> (√3a)/2 = 3x

=> x = (√3a)/6

=> OM = (√3a ) /6

=> radius r = (√3a )/ 6 unit

Or, ( √3*side) /6 unit

Note:- so if each side of equilateral triangle is 5 unit, radius of the inscribed circle = 5√3/6 unit. If each side is 4 unit, radius = 4√3/6 unit.

Hope my answer helps you Dear..

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