find the area of equilateral triangle with measure of side area a,b, c by using multiple double integration

if it is equilateral why does it have three different side lengths?

Assume the base is along the x axis.

Assuming the vertex is between the ends of the base and has x-coordinate v, then if the two sides have equations

y = mx+n and y = hx+k, then just integrate

∫[0,v]∫[0,mx+n] dy dx + ∫[v,a]∫[0,hx+k] dy dx

If v is not directly over the base, things get more complicated...

To find the area of an equilateral triangle using multiple double integration, we can consider the triangle as a combination of two right-angled triangles.

Let's assume that the side length of the equilateral triangle is "a".

First, we need to find the equation of one side of the triangle (let's call it hypotenuse) and the limits of integration.

The equation of the hypotenuse can be expressed as:
y = (sqrt(3)/3)x

To find the limits of integration, we need to determine the x-values where the hypotenuse intersects the x-axis. Since the hypotenuse passes through the origin (0,0), these points will be ±(a/2, 0).

For the right-angled triangle bounded by the x-axis, y-axis, and the hypotenuse, we can integrate with respect to x and y to find its area:

Area of one right-angled triangle = ∫[0 to a/2] ∫[0 to (sqrt(3)/3)x] dy dx

By solving the above integral, we will get the area of one right-angled triangle.

Since the equilateral triangle consists of two right-angled triangles, the total area of the equilateral triangle will be:

Area of equilateral triangle = 2 * (Area of one right-angled triangle)

By following these steps and evaluating the integrals, you can find the area of an equilateral triangle using multiple double integration.