# GRE Math : Equilateral Triangles

## Example Questions

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### Example Question #81 : Geometry

Find the perimeter of an equilateral triangle with a height of .

None of the answer choices are correct.

Explanation:

Perimeter is found by adding up all sides of the triangle. All sides in an equilateral triangle are equal, so we need to find the value of just one side to know the values of all sides.

The height of an equilateral triangle divides it into two equal 30:60:90 triangles, which will have side ratios of 1:2:√3. The height here is the √3 ratio, which in this case is equivalent to 8, so to get the length of the other two sides, we put 8 over √3 (8/√3) and 2 * 8/√3 = 16/√3, which is the hypotenuse of our 30:60:90 triangle.

The perimeter is then 3 * 16/√3, or 48/√3.

### Example Question #1 : Equilateral Triangles

If the height of an equilateral triangle is , what is the perimeter?

Explanation:

By having a height in an equilateral triangle, the angle is bisected therefore creating two  triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite  is  and the side of equilateral triangle which is opposite  is .

Cross multiply.

Divide both sides by

Multiply top and bottom by  to get rid of the radical.

Since each side is the same and there are three sides, we just multply the answer by three to get

### Example Question #3 : Equilateral Triangles

If area of equilateral triangle is , what is the perimeter?

Explanation:

The area of an equilateral triangle is .

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

Cross multiply.

The  cancels out and we get .

Then take square root on both sides and we get . Since we have three equal sides, we just multply  by three to get  as the final answer.

### Example Question #4 : Equilateral Triangles

What is the length of a side of an equilateral triangle if the area is ?

Explanation:

The area of an equilateral triangle is .

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

Cross multiply.

The  cancels out and we get .

Then take square root on both sides and we get  as the final answer.

### Example Question #2 : Equilateral Triangles

If the height of the equilateral triangle is , then what is the length of a side of an equilateral triangle?

Explanation:

By having a height in an equilateral triangle, the angle is bisected therefore creating two  triangles.

The height is opposite the angle . We can set-up a proportion.

Side opposite  is  and the side of equilateral triangle which is opposite  is .

Cross multiply.

Divide both sides by

Multiply top and bottom by  to get rid of the radical.

### Example Question #6 : Equilateral Triangles

Find the area of an equilateral triangle when one of its sides equals 4.

4

8

4√3

2√3

4√3

Explanation:

All sides of an equilateral triangle are equal, so all sides of this triangle equal 4.

Area = 1/2 base * height, so we need to calculate the height: this is easy for an equilateral triangle, since you can bisect any such triangle into two identical 30:60:90 triangles.

The ratio of lengths of a 30:60:90 triangle is 1:2:√3. The side of the equilateral triangle is 4, and we divided the base in half when we bisected the triangle, so that give us a length of 2, so our triangle must have sides of 2, 4, and 2√3; thus we have our height.

One of our 30:60:90 triangles will have a base of 2 and a height of 2√3. Half the base is 1, so 1 * 2√3 = 2√3.

We have two of these triangles, since we divided the original triangle, so the total area is 2 * 2√3 = 4√3.

You can also solve for the area of any equilateral triangle by applying the formula (s2√3)/4, where s = the length of any side.

### Example Question #1 : How To Find The Area Of An Equilateral Triangle

What is the area of an equilateral triangle with a base of ?

Explanation:

An equilateral triangle can be considered to be 2 identical 30-60-90 triangles, giving the triangle a height of . From there, use the formula for the area of a triangle:

### Example Question #91 : Geometry

An equilateral triangle is inscribed into a circle of radius 10. What is the area of the triangle?

The answer cannot be determined from the information given.

Explanation:

To solve this equation, first note that a line drawn from the origin to a vertex of the equilateral triangle will bisect the angle of the vertex. Furthermore, the length of this line is equal to the radius:

That this creates in turn is a 30-60-90 right triangle. Recall that the ratio of the sides of a 30-60-90 triangle is given as:

Therefore, the length of the  side can be found to be

This is also one half of the base of the triangle, so the base of the triangle can be found to be:

Furthermore, the length of the  side is:

The vertical section rising from the origin is the length of the radius, which when combined with the shorter section above gives the height of the triangle:

The area of a triangle is given by one half the base times the height, so we can find the answer as follows:

### Example Question #9 : Equilateral Triangles

One side of an equilateral triangle is equal to

Quantity A: The area of the triangle.

Quantity B:

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

The relationship cannot be determined.

Quantity B is greater.

Explanation:

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

The ratio of sides in a  triangle is , and since the triangle is bisected such that the  degree side is , the  degree side, the height of the triangle, must have a length of .

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 , its area is equal to .

If the relation between ratios is hard to visualize, realize that

### Example Question #92 : Geometry

If the area of an equilateral triangle is , what is the height of the triangle?

Explanation:

The area of an equilateral triangle is .

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

Cross multiply.

The  cancels out and we get .

Then take square root on both sides and we get . To find height, we need to realize by drawing a height we create   triangles.

The height is opposite the angle . We can set-up a proportion. Side opposite  is  and the side of equilateral triangle which is opposite  is .

Cross multiply.

Divide both sides by

We can simplify this by factoring out a  to get a final answer of

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