### All GMAT Math Resources

## Example Questions

### Example Question #1 : Equilateral Triangles

If an equilateral triangle has a perimeter of , what is the length of each side?

**Possible Answers:**

Cannot be determined

**Correct answer:**

An equilateral triangle has three equal sides; therefore, to find the length of each side, divide the perimeter by :

### Example Question #2 : Calculating The Length Of The Side Of An Equilateral Triangle

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

**Possible Answers:**

**Correct answer:**

We derive the equation of base of a triangle from the area of a triangle formula:

### Example Question #3 : Equilateral Triangles

The height of an equilateral triangle is . What is the length of side ?

**Possible Answers:**

**Correct answer:**

Similarily, we can use the same formula for the height to find the length of the side of an equilateral triangle, which is given by

, where is the length of the height.

Therefore, the final answer is

.

### Example Question #1 : Equilateral Triangles

Equilateral triangle is inscribed in a circle with radius , what is the length of a side of the triangle?

**Possible Answers:**

**Correct answer:**

Since we are given the radius, we should be able to find the height of the equilateral triangle. Indeed, the center of the circle is at the intersections of the heights of the triangle, and is located away from the edge of a given height.

Therefore 5, the radius of the circle is of the height.

Therefore, the height must be .

From here, we can use the formula for the height of the equilateral triangle , where is the length of the height and is the length of a side of the equilateral triangle.

Therefore, , then ** **is the final answer.

### Example Question #1 : Equilateral Triangles

Three straight sticks are gathered of exactly equal length. They are placed end to end on the ground to form a triangle. If the area of the triangle they form is 1.732 square feet. What is the length in feet of each stick?

**Possible Answers:**

**Correct answer:**

Let be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side is

So solving

we get .

**Alternative Solution:**

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height, and . Letting stand for the unknown height, we solve

solving for we get

The area for any triangle is the base times the height divided by 2. So

or .

### Example Question #2 : Calculating The Area Of An Equilateral Triangle

If an equilateral triangle has a side length of and a height of , what is the area of the given triangle?

**Possible Answers:**

**Correct answer:**

To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

### Example Question #3 : Calculating The Area Of An Equilateral Triangle

Triangle is an equilateral triangle with side length . What is the area of the triangle?

**Possible Answers:**

**Correct answer:**

The area of an equilateral triangle is given by the following formula:

, where is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have or , which is the final answer.

### Example Question #4 : Calculating The Area Of An Equilateral Triangle

is an equilateral triangle inscribed in a cirlce with radius . What is the area of the triangle ?

**Possible Answers:**

**Correct answer:**

Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is of the length of the height from any vertex.

Therefore, the height is where is the length of the height of the triangle. Therefore .

We can now plug in this value in the formula of the height of an equilateral triangle, where is the length of the side of the triangle.

Therefore, or .

Now we should plug in this value into the formula for the area of an equilateral triangle where is the value of the area of the equilateral triangle. Therefore , which is our final answer.

### Example Question #5 : Calculating The Area Of An Equilateral Triangle

A given equilateral triangle has a side length and a height . What is the area of the triangle?

**Possible Answers:**

Not enough information provided

**Correct answer:**

For a given equilateral triangle with a side length and a height , the area is

. Plugging in the values provided:

### Example Question #6 : Calculating The Area Of An Equilateral Triangle

A given right triangle has a base length and a height . What is the area of the triangle?

**Possible Answers:**

Not enough information to solve

**Correct answer:**

For a given right triangle with a side length and a height , the area is

. Plugging in the values provided:

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