Area of circle which is inscribed in an equilateral triangle?

The area of a circle inscribed inside an equilateral triangle is found using the mathematical formula πa²/12, where 'a' is the side length of the triangle.

Syntax

area = ? × a² / 12

Formula Derivation

Let's see how this formula is derived −

Step 1: Find the radius of the inscribed circle using the formula:

Radius = Area of triangle / Semi-perimeter of triangle

Step 2: For an equilateral triangle with side 'a':

• Area of triangle = (√3)a²/4
• Semi-perimeter = 3a/2

Step 3: Calculate the radius:

Radius = ((√3)a²/4) / (3a/2) = a/(2√3)

Step 4: Find the area of the inscribed circle:

Area = πr² = π × (a/(2√3))² = πa²/12

Example: Calculate Circle Area

This example calculates the area of a circle inscribed in an equilateral triangle with side length 5 −

#include <stdio.h>

int main() {
    float a = 5.0;
    float pi = 3.14159;
    float area = (pi * a * a) / 12;
    
    printf("Side of equilateral triangle: %.2f<br>", a);
    printf("Area of inscribed circle: %.6f<br>", area);
    
    return 0;
}

Side of equilateral triangle: 5.00
Area of inscribed circle: 6.544975

Example: Interactive Calculation

This example demonstrates the calculation for different side lengths −

#include <stdio.h>

int main() {
    float sides[] = {3.0, 6.0, 10.0};
    float pi = 3.14159;
    int n = sizeof(sides) / sizeof(sides[0]);
    
    printf("Side Length\tCircle Area<br>");
    printf("------------------------<br>");
    
    for(int i = 0; i < n; i++) {
        float area = (pi * sides[i] * sides[i]) / 12;
        printf("%.2f\t\t%.6f<br>", sides[i], area);
    }
    
    return 0;
}

Side Length	Circle Area
------------------------
3.00		2.356185
6.00		9.424779
10.00		26.179939

Key Points

• The inscribed circle (incircle) touches all three sides of the triangle
• For any equilateral triangle, the ratio of inscribed circle area to triangle area is π/(3√3)
• The radius of the inscribed circle is always a/(2√3) where 'a' is the side length

Conclusion

The area of a circle inscribed in an equilateral triangle follows the formula πa²/12. This relationship is derived from the geometric properties of equilateral triangles and provides a direct way to calculate the inscribed circle's area given the triangle's side length.