Question

what is the ratio of the radius of circumcircle to the radius of incircle of an equilateral triangle​

Answer 1

Radius of circumcircle of a triangle =

$\frac{abc}{4 \times area}$

Where, a, b and c are sides of the triangle.

Now, radius of incircle of a triangle =

$\frac{area}{s}$

where, s = semiperimeter.

Now for an equilateral triangle, sides are equal. Let the side be a

Hence, its semiperimeter = 3a/2

and area = √3/4 × a²

So, circumradius =

$\frac{a \times a \times a}{4 \times \frac{ \sqrt{3} }{4} {a}^{2} } \\ \\ = \frac{ {a}^{3} }{ \sqrt{3} {a}^{2} } \\ \\ = \frac{a}{ \sqrt{3} } \\ \\ = \frac{ \sqrt{3} }{3} a$

Now, inradius of the equilateral triangle =

$\frac{ \frac{ \sqrt{3} }{4} {a}^{2} }{ \frac{3a}{2} } \\ \\ = \frac{ \sqrt{3} }{4} {a}^{2} \times \frac{2}{3a} \\ \\ = > \frac{a}{2 \sqrt{3} } \\ \\ = \frac{ \sqrt{3}a }{6}$

So ratio of circumradius to inradius =

$\frac{ \frac{ \sqrt{3}a }{3} }{ \frac{ \sqrt{3}a }{6} } \\ \\ = \frac{ \sqrt{3} }{3} a \times \frac{6}{ \sqrt{3}a } \\ \\ = \frac{2}{1}$

Hence, ratio of circumradius to inradius = 2 : 1

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