Area of Reuleaux Triangle?

A Reuleaux Triangle is a special geometric shape formed by the intersection of three circles. It is constructed using an equilateral triangle as a base, where each side of the triangle becomes the radius of a circle centered at the opposite vertex. The area calculation involves three circular sectors minus the overlapping equilateral triangle areas.

Syntax

float areaReuleaux(float h);
// where h is the height of the equilateral triangle

Formula

The area of a Reuleaux Triangle is calculated using the formula −

Area = (? - ?3) × h² / 2

Where:

• h is the height of the equilateral triangle
• ? ? 3.14159
• ?3 ? 1.732

Example

This program calculates the area of a Reuleaux Triangle given the height of the inner equilateral triangle −

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

float areaReuleaux(float h) {
    if (h < 0) // if h is negative it is invalid
        return -1;
    float area = ((3.14159 - sqrt(3)) * h * h) / 2;
    return area;
}

int main() {
    float height = 6;
    printf("Height of triangle: %.2f
", height);
    printf("Area of Reuleaux Triangle: %.4f
", areaReuleaux(height));
    return 0;
}

Height of triangle: 6.00
Area of Reuleaux Triangle: 25.3701

How It Works

The calculation involves these steps:

• Three circular sectors each contribute ?h²/6 to the total area
• The equilateral triangle area ?3h²/4 is counted three times and must be subtracted twice
• Final formula: 3(?h²/6) - 2(?3h²/4) = (? - ?3)h²/2

Conclusion

The Reuleaux Triangle area formula (? - ?3)h²/2 efficiently calculates the area using only the height of the inner equilateral triangle. This constant-width curve has applications in engineering and mechanical design.