GRE Math : Equilateral Triangles
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Example Question #95 : Geometry
Quantity A: The height of an equilateral triangle with an area of
Quantity B:
Which of the following is true?
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined.
Quantity A is greater.
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 #2 : How To Find The Height Of An Equilateral Triangle
Quantity A: The height of an equilateral triangle with perimeter of 
Quantity B:
Which of the following is true?
The two quantities are equal.
Quantity B is larger.
The relationship cannot be determined.
Quantity A 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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