Area of a circle inscribed in a rectangle which is inscribed in a semicircle?

Let us consider a semicircle with radius R. A rectangle of length l and breadth b is inscribed in that semicircle. Now a circle with radius r is inscribed in the rectangle. We need to find the area of the inner circle.

As we know, the largest rectangle that can be inscribed within a semicircle has the following relationship between its dimensions and the semicircle's radius −

Mathematical Relationships

For the rectangle inscribed in the semicircle:

• Length: l = R?2
• Breadth: b = R/?2

The largest circle that can be inscribed within the rectangle has radius equal to half the smaller dimension:

• Inscribed circle radius: r = b/2 = R/(2?2)

Therefore, the area of the inscribed circle is:

• Area: ?r² = ? × (R/(2?2))² = ?R²/8

Syntax

float innerCircleArea(float R);

Example

Here's a C program to calculate the area of a circle inscribed in a rectangle which is inscribed in a semicircle −

#include <stdio.h>
#include <math.h>

float innerCircleArea(float R) {
    float radius = R / (2 * sqrt(2));
    return 3.14159 * radius * radius;
}

int main() {
    float semicircleRadius = 12.0;
    float area = innerCircleArea(semicircleRadius);
    
    printf("Semicircle radius: %.2f<br>", semicircleRadius);
    printf("Inscribed circle area: %.3f<br>", area);
    
    return 0;
}

Output

Semicircle radius: 12.00
Inscribed circle area: 56.549

Key Points

• The inscribed circle's radius is R/(2?2) where R is the semicircle's radius.
• The area formula simplifies to ?R²/8.
• This represents the maximum area circle that can fit inside the optimal rectangle.

Conclusion

The area of a circle inscribed in a rectangle inscribed in a semicircle is ?R²/8, where R is the semicircle's radius. This geometric relationship provides an elegant solution for nested geometric shapes.