### All SSAT Upper Level Math Resources

## Example Questions

### Example Question #1 : Equilateral Triangles

An equilateral triangle is circumscribed about a circle of radius 16. Give the area of the triangle.

**Possible Answers:**

The correct answer is not among the other choices.

**Correct answer:**

The circle and triangle referenced are below, along with a radius to one side and a segment to one vertex:

is a 30-60-90 triangle, so

is one-half of a side of the triangle, so the sidelength is . The area of the triangle is

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

In the above diagram, is equilateral. Give its area.

**Possible Answers:**

The correct answer is not among the other responses.

**Correct answer:**

The correct answer is not among the other responses.

The interior angles of an equilateral triangle all measure 60 degrees, so, by the 30-60-90 Theorem,

Also, is the midpoint of , so ; this is the base.

The area of this triangle is half the product of the base and the height :

This answer is not among the given choices.

### Example Question #3 : How To Find The Area Of An Equilateral Triangle

The perimeter of an equilateral triangle is . Give its area.

**Possible Answers:**

**Correct answer:**

An equilateral triangle with perimeter has three congruent sides of length

The area of this triangle is

### Example Question #4 : How To Find The Area Of An Equilateral Triangle

Hexagon is regular and has perimeter 72. is constructed. What is its area?

**Possible Answers:**

**Correct answer:**

Since the perimeter of the (six-congruent-sided) regular hexagon is 72, each side has length one sixth this, or 12.

The figure described is given below, with a perpendicular segment drawn from to side :

Each angle of a regular hexagon measures . Therefore, , and is a 30-60-90 triangle.

By the 30-60-90 Theorem,

and

.

is equilateral, and is its sidelength, making its area

### Example Question #5 : How To Find The Area Of An Equilateral Triangle

The perimeter of an equilateral triangle is . Give its area.

**Possible Answers:**

**Correct answer:**

An equilateral triangle with perimeter 54 has three congruent sides of length

The area of this triangle is

### Example Question #6 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is inscribed inside a circle of radius . Give the area of the triangle.

**Possible Answers:**

**Correct answer:**

The trick is to know that the circumscribed circle, or the *circumcircle*, has as its center the intersection of the three altitudes of the triangle, and that this center, or *circumcenter*, divides each altitude into two segments, one twice the length of the other - the longer one being a radius. Because of this, we can construct the following:

Each of the six smaller triangles is a 30-60-90 triangle, and all six are congruent.

We will find the area of , and multiply it by 6.

By the 30-60-90 Theorem, , so the area of is

.

Six times this - - is the area of .

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

An equilateral triangle has side lengths of . What is the area of this triangle?

**Possible Answers:**

**Correct answer:**

The area of an equilateral triangle can be found using this formula:

Using , we can find the area of the equilateral triangle.

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