Question

What is the area of an equilateral triangle inscribed in a circle?

Answer 1

Let ABC equatorial triangle inscribed in the circle with radius r

Applying law of sine to the triangle OBC, we get

`a/sin60=r/sin30=>a=r*sin60/sin30=>a=sqrt3*r`

Now the area of the inscribed triangle is

`A=1/2*AM*ΒC`

Now `AM=AO+OM=r+r*sin30=3/2*r`

and `ΒC=a=sqrt3*r`

Finally

`A=1/2*(3/2*r)*(sqrt3*r)=1/4*3*sqrt3*r^2`