ACT Math : How to find the length of the side of an equilateral triangle
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Example Question #161 : Plane Geometry
The height of an equilateral triangle is 
From the given information, we can draw the following picture:
Since this is an equilateral triangle, all three of its angles will be congruent. All three of its sides will also be congruent.
The dotted line, representing the height, bisects not only the base of the triangle but also the apical angle. This means that the two smaller right triangles created by the dotted line have bases that are half the side length of the equilateral triangle with apical angle measures of 
This creates two smaller 30/60/90 special right triangles. This problem can be solved in one of two ways:
1. You can use the derived side ratio for 30/60/90 triangles, 
OR
2. You can use trig functions to solve for the length of one of the equilateral triangle's sides.
Using trig functions, we can use 
Going back to SOH CAH TOA, we can determine that we will be using sine because we have the information available for the side opposite ("O") of our angle of interest and are looking for the hypotenuse ("H"). This means we will be using the "SOH" part of SOH CAH TOA—sine.
Substituting in the problem's values, we can solve for the length of the hypotenuse of one of the two smaller triangles, which is the length of one side of the equilateral triangle:
Remember that the problem asks for the triangle's perimeter. That means that we need to multiply the length of one of its edges by three to find the final answer:
Example Question #2 : How To Find The Length Of The Side Of An Equilateral Triangle
If an equilateral triangle has a height of 
For any triangle, 
We know our height is 
Therefore 
Example Question #1 : How To Find The Length Of The Side Of An Equilateral Triangle
An equilateral triangle has an area of 
Since this triangle is equilateral, all of the sides (including the base) are equal.
Use the formula for area of a triangle to solve for the base:
In this case:
So the base is equal to
Example Question #1 : How To Find The Length Of The Side Of An Equilateral Triangle
An equilateral triangle has a height of 
Since this is an equilateral triangle, all of the sides and all of the angles would be equal (60 degrees each since the 3 angles must add up to 180 degrees).
1. The height splits the triangle's base in half and creates two right triangles. Create expressions for the length of the two unknown sides in this new triangle:
The value we are looking for is the side length so we'll make that
The base of the new triangle is half the length of a side so we'll make that
2. Use the Pythagorean Theorem to find the value of 
In this case:
3. Multiply the side length you found by 3 to get the perimeter:
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