By definition, an equilateral triangle is one whose sides all possess equal lengths. As is the case with any triangle or geometric shape, you can use a simple formula to calculate the perimeter of an equilateral triangle. In order to apply the standard formula for determining a triangle's area to an equilateral triangle, though, you must manually calculate its height. Another approach is to use a different formula for area intended especially for equilateral triangles.
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Difficulty:
Moderately Easy
Instructions
1
Compute your equilateral triangle's perimeter by adding the length of all its sides together--or, since the sides of an equilateral triangle are equal, simply multiple one side's length by 3. For example, if your equilateral triangle's sides are three units in length, its perimeter will be 9 units.
2
In order to calculate the area of your equilateral triangle using the standard formula for a triangle's area, first use the Pythagorean theorem (a^2 + b^2 = c^2) to calculate your equilateral triangle's height. For example, think of an equilateral triangle with a side length of 3 units as two right triangles with 1.5-unit bases. Applying the Pythagorean theorem, you will see that a^2 + 1.5^2 = 3^2, which means that a^2 + 2.25 = 9, so a^2 = 6.75. Performing a square root to find the value of "a" (height), you will find that your triangle has a height of 2.6 units.
3
Apply the standard formula for the area of a triangle, a = 1/2 (b x h). For the equilateral triangle with sides 3 units in length, this will be 1/2 (3 x 2.6), or 3.9 square units.
4
Use the special formula for the area of equilateral triangles to check your answer. For equilateral triangles, the special area formula is a = (s^2 * 1.73)/4, where s = side length and the value "1.73" represents the square root of 3 (a constant in the formula because an equilateral triangle has 3 sides, not because this example has a side length of 3 units). You will see that for a triangle with a side length of 3, this formula yields an identical area to the other one, 3.9 square units.
Tips & Warnings
In order to use the Pythagorean theorem, which only works for right triangles, you must consider your equilateral triangle as two right triangles of the same height, with bases equal to one-half your equilateral triangle's base. As far as the theorem itself is concerned, keep in mind that "a" and "b" represent "height" and "base" respectively, while "c" represents the hypotenuse. In other words, you can think of the "height" as the square root of the difference of the hypotenuse squared and the base squared, or a = the square root of (c^2 - b^2).
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