Inscribed Circle in a Right Triangle

Problem

A right triangle has legs $6$ and $8$, and a circle is inscribed in it; find the radius of the circle and the shaded area between the triangle and the circle.

Step 1: Find the hypotenuse

Using the Pythagorean theorem,

$$c^2 = 6^2 + 8^2 = 36 + 64 = 100,$$

so the hypotenuse is

$$c = 10.$$

Step 2: Use the inradius formula

For a right triangle, the inradius is

$$r = \frac{a+b-c}{2}.$$

Substituting $a=6$, $b=8$, and $c=10$ gives

$$r = \frac{6+8-10}{2} = \frac{4}{2} = 2.$$

So the circle's radius is $2$.

Step 3: Find the shaded area

The shaded region is the triangle area minus the circle area.

The triangle area is

$$\frac{1}{2}\cdot 6 \cdot 8 = 24.$$

The circle area is

$$\pi r^2 = \pi(2^2) = 4\pi.$$

So the shaded area is

$$24 - 4\pi,$$

which is about $11.43$ square units.

Answer

The radius is $2$, and the shaded area is $24 - 4\pi$.