GRE Math : How to find the height of an equilateral triangle

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Example Question #32 : Triangles

One side of an equilateral triangle is equal to

Quantity A: The area of the triangle.

Quantity B: $\frac{1}{2}$

Possible Answers:

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

The relationship cannot be determined.

Correct answer:

Quantity B is greater.

Explanation:

To find the area of an equilateral triangle, notice that it can be divided into two 30-60-90 triangles:

Equilateral triangle

The ratio of sides in a 30-60-90 triangle is $1-\sqrt{3}-2$ , and since the triangle is bisected such that the degree side is $\frac{s}{2}$ , the degree side, the height of the triangle, must have a length of $\frac{\sqrt{3}}{2}s$ .

The formula for the area of the triangle is given as:

$Area =\frac{1}{2}base*height$

So the area of an equilateral triangle can be written in term of the lengths of its sides as:

$\frac{1}{2}s\frac{\sqrt{3}s}{2}=\frac{\sqrt{3}}{4}s^2$

For this particular triangle, since s=1 , its area is equal to $\frac{\sqrt{3}}{4}$ .

$\frac{\sqrt{3}}{4}<\frac{1}{2}$

If the relation between ratios is hard to visualize, realize that $\frac{\sqrt{4}}{4}=\frac{2}{4}=\frac{1}{2}$

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Example Question #2 : How To Find The Height Of An Equilateral Triangle

If the area of an equilateral triangle is $6\sqrt{3}$ , what is the height of the triangle?

Possible Answers:

$2\sqrt{6}$

$\sqrt{6}$

$2\sqrt{3}$

$\frac{3\sqrt{2}}{2}$

$3\sqrt{2}$

Correct answer:

$3\sqrt{2}$

Explanation:

The area of an equilateral triangle is $\frac{s^2\sqrt{3}}{4}$ .

So let's set-up an equation to solve for .

$\frac{s^2\sqrt{3}}{4}=6\sqrt{3}$ Cross multiply.

$s^2\sqrt{3}=24\sqrt{3}$

The $\sqrt{3}$ cancels out and we get s^2=24 .

Then take square root on both sides and we get $2\sqrt{6}$ . To find height, we need to realize by drawing a height we create 30-60-90 triangles.

The height is opposite the angle . We can set-up a proportion. Side opposite is $\sqrt{3}$ and the side of equilateral triangle which is opposite is .

$\frac{h}{\sqrt{3}}=\frac{2\sqrt{6}}{2}$ Cross multiply.

$2\sqrt{18}=2h$ Divide both sides by

$h=\sqrt{18}$

We can simplify this by factoring out a $\sqrt{9}$ to get a final answer of $3\sqrt{2}$ .

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Example Question #95 : Geometry

Quantity A: The height of an equilateral triangle with an area of $8\sqrt{3}$

Quantity B:

Which of the following is true?

Possible Answers:

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined.

Quantity A is greater.

Correct answer:

Quantity B is greater.

Explanation:

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 $\sqrt{3}$ .

Consider the equilateral triangle:

Equilateral8

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:

Equilateral8 2

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:

$\frac{1}{\sqrt{3}}$

We can also say, given our figure, that the following equivalence must hold:

$\frac{1}{\sqrt{3}} = \frac{4}{x}$

Solving for , we get:

$x = 4\sqrt{3}$

Now, since $\sqrt{3} < \sqrt{4}$ , we know that $\sqrt{3}$ must be smaller than . This means that $4\sqrt{3} < 4 * 2$ or $4\sqrt{3} < 8$ . Quantity B is larger than quantity A.

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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?

Possible Answers:

The two quantities are equal.

Quantity B is larger.

The relationship cannot be determined.

Quantity A is larger.

Correct answer:

Quantity B is larger.

Explanation:

If the perimeter of our equilateral triangle is , each of its sides must be $\frac{27}{3}$ or . This gives us the following figure:

Equilateral9

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:

Equilateral9 2

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:

$\frac{1}{\sqrt{3}}$

Therefore, we can also say, given our figure, that the following equivalence must hold:

$\frac{1}{\sqrt{3}} = \frac{4.5}{x}$

Solving for , we get:

$x = 4.5\sqrt{3}$

Now, since $\sqrt{3} < \sqrt{4}$ , we know that $\sqrt{3}$ must be smaller than . This means that $4.5\sqrt{3} < 4.5 * 2$ or $4.5\sqrt{3} < 9$

Therefore, quantity B is larger than quantity A.

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GRE Math : How to find the height of an equilateral triangle

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #32 : Triangles

Example Question #2 : How To Find The Height Of An Equilateral Triangle

Example Question #95 : Geometry

Example Question #2 : How To Find The Height Of An Equilateral Triangle

All GRE Math Resources

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