Area of Circumcircle of a Right Angled Triangle?

The area of circumcircle of a right-angled triangle can be calculated when the hypotenuse (H) of the triangle is given using the formula ?H²/4. This formula is derived from the fact that in a right-angled triangle, the hypotenuse is the diameter of the circumcircle.

The circumcircle passes through all three vertices of the triangle. For a right-angled triangle, the hypotenuse becomes the diameter because the angle inscribed in a semicircle is always 90°. Since the area of a circle is ?r², and diameter d = 2r, we can write the area as ?d²/4, where d is replaced by the hypotenuse H.

Syntax

area = (? * H * H) / 4

Example

Let's calculate the area of circumcircle for a right-angled triangle with hypotenuse = 14 −

#include <stdio.h>

int main() {
    float H = 14.0;
    float pi = 3.14159;
    float area = (pi * H * H) / 4;
    
    printf("Hypotenuse: %.1f<br>", H);
    printf("Area of circumcircle: %.2f<br>", area);
    return 0;
}

Hypotenuse: 14.0
Area of circumcircle: 153.94

Key Points

• In a right-angled triangle, the hypotenuse is always the diameter of the circumcircle
• The formula ?H²/4 directly gives the area without needing the radius
• This property is unique to right-angled triangles due to Thales' theorem

Conclusion

The circumcircle area of a right-angled triangle is easily calculated using ?H²/4, where H is the hypotenuse. This elegant formula exploits the geometric property that makes the hypotenuse the circle's diameter.