SSAT Upper Level Math : How to find the area of an equilateral triangle

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

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

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:

Explanation:

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

Equilateral

 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

Equilateral

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.

Explanation:

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:

Explanation:

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:

Explanation:

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 :

Hexagon

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:

Explanation:

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:

Explanation:

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:

Equilateral

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 #7 : 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:

Explanation:

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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