# Calculating the area of equilateral triangle using calculus

In summary, the conversation discusses different methods for calculating the area of an equilateral triangle using calculus. The first method involves breaking the triangle into smaller shapes and finding their areas, while the second method suggests directly integrating the area of the triangle. However, there is a discrepancy in the result obtained using the first method and the expected result. The conversation then suggests finding the equation of the line for the triangle in order to accurately calculate its area.

## Homework Statement

Calculating the area of equilateral triangle using calculus.

## The Attempt at a Solution

The area of the triangle is the area of the circle minus 3 times the area of the sector shown in (light blue). So, the target is to calculate the pink area first.
##y=\sqrt{r^2-x^2}##
The pink area is ##\int_{0}^{x_0} \sqrt{r^2-x^2} dx## = ##\int_{0}^{x_0} r\sqrt{1-\frac{x^2}{r^2}} dx## Putting ##x=r sin a## and doing the usual math with integration from 0 to ##\pi/6## led me to;
##r^2\int_{0}^{\pi/6} cos^2 a da##=##\frac{r^2}{2}\int_{0}^{\pi/6} cos (2a+1) da##=##\frac{r^2}{2} \left[\frac{sin2a}{2}+a\right]_0^{\pi/6}##=##\frac{r^2}{2} [\frac{1}{2}\frac{\sqrt 3}{2}+\pi/6]##=##\frac{r^2 \sqrt 3}{8} +\frac{\pi}{12}##
The blue area is then ##2 (\frac{\pi r^2}{4}-\frac{r^2 \sqrt 3}{8} -\frac{\pi}{12})##=##\frac{\pi r^2}{2}-\frac{r^2 \sqrt 3}{4} -\frac{\pi}{6})## and then the area of triangle is ##\pi r^2-3(\frac{\pi r^2}{2}-\frac{r^2 \sqrt 3}{4} -\frac{\pi}{6})##
This will not give the correct result of ##\frac{3\sqrt3 r^2}{4}##

Since you know the angle a = pi/6, you should be able to calculate the x value where the blue area starts, then just integrate sqrt(r2 - x2) from there to r. That gets you the top half of the blue area.

It's so much easier to integrate the area of the triangle directly.

Skins and scottdave
willem2 said:
It's so much easier to integrate the area of the triangle directly.
That too. With what he has figured, it should be easy to figure the equation of the line for the triangle.

## What is an equilateral triangle?

An equilateral triangle is a type of triangle in which all three sides are equal in length.

## What is calculus?

Calculus is a branch of mathematics that deals with rates of change and the calculation of areas and volumes of shapes.

## How do you calculate the area of an equilateral triangle using calculus?

The formula for calculating the area of an equilateral triangle using calculus is A = √3/4 * s^2, where A is the area and s is the length of one side.

## Why is calculus used to calculate the area of an equilateral triangle?

Calculus is used because it allows us to find the exact area of curves and irregular shapes, such as an equilateral triangle, by breaking them down into infinitely small, straight sections and using mathematical formulas to find the overall area.

## What are the other methods for calculating the area of an equilateral triangle?

Other methods for calculating the area of an equilateral triangle include using the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Additionally, the area can also be calculated by dividing the triangle into two right triangles and using the formula for finding the area of a right triangle, A = 1/2 * base * height.

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