The largest circle is inscribed in a segment of a circle of radius R, bounded by an arc of 90 °

The largest circle is inscribed in a segment of a circle of radius R, bounded by an arc of 90 ° and a chord that contracts it. Find its radius.

The degree measure of the arc is equal to the central angle and it is 90 °. So we will consider a right-angled triangle with legs equal to R:
a = b = R.

Let’s calculate the hypotenuse of this triangle:
c² = a² + b² = R² + R² = 2 * R².
c = √2 * R.

Find the height of the triangle:
a * b / c = R * R / (√2 * R) = R / √2.

Determine the height of the circle segment:
h = R – R / √2.

Find the radius of the circle inscribed in the segment:
r = h / 2 = (R – R / √2) / 2 = R * (√2 – 1) / (2 * √2) = R * (2 – √2) / 4.