### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : Equilateral Triangles

The length of one side of an equilateral triangle is 6 inches. Give the area of the triangle.

**Possible Answers:**

**Correct answer:**

,

where and are the lengths of two sides of the triangle and is the angle measure.

In an equilateral triangle, all of the sides have the same length, and all three angles are always .

### Example Question #2 : 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

, so

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

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

**Possible Answers:**

The correct answer is not among the other four choices.

**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

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

**Possible Answers:**

**Correct answer:**

An equilateral triangle with perimeter has three congruent sides of length . Substitute this for in the following area formula:

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

An equilateral triangle is inscribed inside a circle of radius 8. Give its area.

**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 #6 : 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 36 has three congruent sides of length

The area of this triangle is

### Example Question #111 : Geometry

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

**Possible Answers:**

**Correct answer:**

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 :

### Example Question #8 : How To Find The Area Of An Equilateral Triangle

Refer to the above figure. The shaded region is a semicircle with area . Give the area of .

**Possible Answers:**

**Correct answer:**

Given the radius of a semicircle, its area can be calculated using the formula

.

Substituting :

The diameter of this semicircle is twice this, which is ; this is also the length of .

has two angles of degree measure 60; its third angle must also have measure 60, making an equilateral triangle with sidelength . Substitute this in the area formula: