What is the area of the largest triangle that can be fit in a circle of radius 12?

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

An equilateral triangle that can fit in a circle has the largest area of all triangles that can be placed in a circle.

Now for an equilateral triangle with sides a, the area is given by (sqrt 3 / 4)*a^2

The radius of the circumscribed circle is a / sqrt 3.

Now we have a circle of radius 12.

Therefore a / sqrt 3 = 12

=> a = 12 * sqrt 3

Now the area of a triangle with side 12/ sqrt 3 is

=> (sqrt 3 / 4)*(12 * sqrt 3) ^2

=> (sqrt 3 / 4)* (144 * 3)

=> (144*sqrt 3* 3/ 4)

=> 108*sqrt 3

The area of the largest triangle that can be inscribed in a circle of radius 12 is 108*sqrt 3.

neela | High School Teacher | (Level 3) Valedictorian

Posted on

We know that the largest area that can be formed in a circle is the equilateral triangle.

We know the area of  a triangle = (1/2)bcsinA = (1/2) a^2 sin 60, as in case of an equilateral triangle a= b=c .

Therefore Area of the largest triangle = (1/2)a^2 * (sqrt3)/2 = (a^2)(sqrt3)/4.

Also  a/sin60 = 2R

So a =2*12sin 60

So  a = 2*12*(sqrt3)/2 = 12 sqrt(3).

Therefore Area of the largest triangle ={12 sqrt3}^2 (sqrt3)/4 = 12^2*3* sqr3/4 = 187.06 sq units.