A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(3pi)/8 #, and the triangle's area is #3 #. What is the area of the triangle's incircle?

Question

A triangle has vertices A, B, and C.  Vertex A has an angle of #pi/12 #, vertex B has an angle of #(3pi)/8  #, and the triangle's area is #3 #.  What is the area of the triangle's incircle?

Aurora Ayala · Accepted Answer

Area of the incircle is 1.7232 sq units

The incentre of a triangle is the point of intersection of the angle bisectors of angles of the triangle. An incentre is also the centre of the circle touching all the sides of the triangle.

Note:
Angle bisector divides the oppsoite sides in the ratio of remaining sides i.e. BD/DC = AB/AC = c/b.

Incentre divides the angle bisectors in the ratio (b+c):a, (c+a):b and (a+b):c

James Atkins · Answer

undefined To find the area of the triangle's incircle, you can use the formula:

Area of incircle = s * r,

where 's' is the semi-perimeter of the triangle and 'r' is the radius of the incircle.

First, calculate the semi-perimeter 's' using the formula:

s = (a + b + c) / 2,

where 'a', 'b', and 'c' are the lengths of the sides of the triangle.

Next, use Heron's formula to find the area of the triangle:

Area = √(s * (s - a) * (s - b) * (s - c)).

Once you have the area of the triangle, you can find the radius 'r' of the incircle using the formula:

r = Area / s.

Then, multiply the semi-perimeter 's' by the radius 'r' to find the area of the incircle.