What is the area of a circle within a rectangle inscribed in a semicircle?

Let us consider a given semicircle. Its radius is R. A rectangle of length l and width b is inscribed in the semicircle. Now, a circle of radius r is inscribed in the rectangle. We must find the area of the inner circle.

We know that the length of the largest rectangle inscribed in a semicircle is l and the width is b. Then the equations of l and b are as follows-

Now, the radius of the largest circle that can be inscribed in the rectangle is r as follows-

Example

#include  #include  using namespace std; float innerCircleArea(float R){ return 3.1415 * pow(R / (2 * sqrt(2)), 2); } int main() { float rad = 12.0f; cout << "Area: " << innerCircleArea(rad); }

Output

Area: 56.547

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