# GRE Math : Diameter

## Example Questions

### Example Question #1 : Diameter

The formula to find the radius of the largest circle that can fit in an equilateral triangle is , where  is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of  cm?

cm

cm

cm

cm

cm

Explanation:

The diameter is

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

.

Substitute in 5  for S and solve. Diameter =  = 2.89 cm

### Example Question #2 : Diameter

Quantity A: The diameter of a circle with area of

Quantity B: The diameter of a circle with circumference of

Which of the following is true?

Quantity B is larger.

The relationship of the quantities cannot be determined.

Quantity A is larger.

Both quantities are equal.

Quantity B is larger.

Explanation:

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

Therefore, if . Therefore, Quantity B is larger.

### Example Question #1 : How To Find The Length Of The Diameter

Quantity A: The diameter of a circle with area of

Quantity B: The diameter of a circle with circumference of

Which of the following is true?

The relationship between the quantities cannot be determined.

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

Quantity B is larger.

Explanation:

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

We know that the area is . Therefore,

Divide both sides by :

Therefore, .  Since , we know:

Quantity B

This is very easy.  Recall that:

Therefore, if .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that  is . Therefore, .  This means that . Therefore, Quantity B is larger.

### Example Question #4 : Diameter

A circle with an area of  is divided into sectors with areas in a ratio of . What is the area of the largest sector?

Explanation:

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

or

The largest sector thus has an area equal to

### Example Question #5 : Diameter

A rectangle is inscribed inside of a circle such that every corner touches the edge of the circle. If the area of the rectangle is  and the perimeter of the rectangle is , what is the area of the circle in inches squared?

Explanation:

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for ,  or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

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