A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/8 #, and the triangle's area is #4 #. 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 #pi/8  #, and the triangle's area is #4 #.  What is the area of the triangle's incircle?

Sadie Allen · Accepted Answer

#color(purple)("Area of incircle " A_r = pi r^2 = 0.00166 " sq units"#

The distances from the incenter to each side are equal to the inscribed circle's radius. The area of the triangle is equal to 12×r×(the triangle's perimeter), 1 2 × r × ( the triangle's perimeter ) , where r is the inscribed circle's radius.

First to find the perimeter of the triangle.

#hat A = pi/12, hat B = pi/8, hat C = (19pi)/24, A_t = 4#

Area of triangle #A_t = (1/2)ab sin C = (1/2)bc sin A = (1/2)ca sin B#

#ab = (4 * 2) / sin ((19)/24) = 13.14#

Similarly, #bc = (4 * 2) / sin (pi/12) = 30.91#

#ca = (4 * 2) / sin (pi/8) = 20.91#

#ab * bc * ca = (abc)^2 = 13.14 * 30.91 * 20.91#

#abc = sqrt(13.14 * 30.91 * 20.91)#

#a = (abc) / (bc) = sqrt(13.14 * 30.91 * 20.91) / 30.91 = 2.98#

Likewise, #b = (abc) / (ca) = sqrt(13.14 * 30.91 * 20.91) / 20.91 = 4.41#

#c = (abc) / (ab) = sqrt(13.14 * 30.91 * 20.91) / 13.14 = 7.01#

#"Perimeter of the triangle " p = a + b + c = 2.98 + 4.41 + 7.01 = 14.4#

#"Radius of incircle " r = A_t / (12 * p) = 4 / (12 * 14.4) = 0.023#

#color(purple)("Area of incircle " A_r = pi r^2 = pi * 0.023^2 = 0.00166#

Maverick Smith · Answer

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

$$ \text{Area of incircle} = \pi \times r^2 $$

where $ r $ is the radius of the incircle. The radius $ r $ can be found using the formula:

$$ r = \frac{\text{Area of triangle}}{s} $$

where $ s $ is the semi-perimeter of the triangle, given by:

$$ s = \frac{a + b + c}{2} $$

Given the angles of the triangle, you can use the law of sines to find the lengths of the sides. The law of sines states:

$$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$

Using this, you can find the lengths of the sides $ a $, $ b $, and $ c $. Then, use the semi-perimeter to find $ r $, and finally, use $ r $ to find the area of the incircle.