Let $ab = 6$, $bc = 8$, and $ac = 10$. what is the area of the circumcircle of $\triangle abc$ minus the area of the incircle of $\triangle abc$?

The problem is illustrated by the figure shown below.

It is clear that ΔABC is a right triangle because
(a) it is inscribed in a circle,
(b) Its sides satisfy the Pythagorean theorem because
      6² + 8² = 36 + 64 = 100, and 10² = 100.

Therefore, AC is the diameter of the circumscribing circle, and the radius is
 r = 5.

The area of the circumscribing circle is 
A₁ = π(5²) = 25π

The area of the triangle is
A₂ = (1/2) 8*6 = 24

The required area, of the circumcircle minus that of the triangle (shaded area) is
A₁ - A₂ = 25π - 24 = 54.54 square units.

54.54 square units.

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