# GRE Math : How to find the length of the diameter

## Example Questions

### Example Question #1 : How To Find The Length Of The 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 : 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 of the quantities cannot be determined.

Quantity A is larger.

Quantity B 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 #2 : 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 two quantities are equal.

The relationship between the quantities cannot be determined.

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.
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