GRE Math : How to find the height of an equilateral triangle
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Example Questions
Example Question #1 : How To Find The Height Of An Equilateral Triangle
One side of an equilateral triangle is equal to
Quantity A: The area of the triangle.
Quantity B:
The two quantities are equal.
Quantity A is greater.
Quantity B is greater.
The relationship cannot be determined.
Quantity B is greater.
To find the area of an equilateral triangle, notice that it can be divided into two 
The ratio of sides in a 
The formula for the area of the triangle is given as:
So the area of an equilateral triangle can be written in term of the lengths of its sides as:
For this particular triangle, since 
If the relation between ratios is hard to visualize, realize that
Example Question #1 : How To Find The Height Of An Equilateral Triangle
If the area of an equilateral triangle is 
The area of an equilateral triangle is 
So let's set-up an equation to solve for 
The 
Then take square root on both sides and we get 
The height is opposite the angle 
We can simplify this by factoring out a 
Example Question #41 : Triangles
Quantity A: The height of an equilateral triangle with an area of
Quantity B:
Which of the following is true?
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined.
Quantity B is greater.
This problem requires a bit of creative thinking (unless you have memorized the fact that an equilateral triangle always has an area equal to its side length times 
Consider the equilateral triangle:
Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:
This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:
We can also say, given our figure, that the following equivalence must hold:
Solving for 
Now, since 
Example Question #41 : Triangles
Quantity A: The height of an equilateral triangle with perimeter of 
Quantity B:
Which of the following is true?
Quantity A is larger.
The two quantities are equal.
The relationship cannot be determined.
Quantity B is larger.
Quantity B is larger.
If the perimeter of our equilateral triangle is 
Since this kind of triangle is a species of isoceles triangle, we know that we can drop down a height from the top vertex. This will create two equivalent triangles, one of which will look like:
This gives us a 30-60-90 triangle. We know that for such a triangle, the ratio of the side across from the 30-degree angle to the side across from the 60-degree angle is:
Therefore, we can also say, given our figure, that the following equivalence must hold:
Solving for 
Now, since 
Therefore, quantity B is larger than quantity A.
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