A triangle has side lengths of $10,$ $24,$ and $26.$ Let $a$ be the area of the circumcircle. Let $b$ be the area of the incircle. Compute $a - b.$

Step-by-step explanation:

For the inscribed circle

Formula for calculating the radius of an inscribed circle in a triangle r = Area of the triangle/0.5 of its perimeter

Given a triangle of side 10, 24 and 26.

Area of the triangle = 1/2 * base * height

Area of the triangle = 1/2*10*24

Area of the triangle = 120

Perimeter of the triangle = 10+24+26 = 60

radius of the inscribed circle = 120/0.5(60)

r = 120/30

r = 4

Area of the incircle b = πr² = π(4)² = 16π

For the circumcircle

Formula for calculating the radius of a circumcircle outside a triangle

R = abc/4√s(s-a)(s-b)(s-c)

s = a+b+c/2

a, b and c are the length of the 3 sides

s = 10+24+26/2

s = 60/2 = 30

R = 10(24)(26)/4√30(20)(6)(4)

R = 6240/4*120

R = 13

radius of the circumcircle = 13

Area of the circumcircle a = πR² = π(13)² = 169π

a - b = 169π - 16π

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