# GMAT Math : Calculating the ratio of diameter and circumference

## Example Questions

### Example Question #91 : Geometry

For any circle, what is the ratio of its circumference to its diameter?

Explanation:

In order to calculate the ratio of circumference to diameter, we need an equation that involves both variables. The formula for circumference is as follows:

We need to express the radius in terms of diameter. The radius of a circle is half of its diameter, so we can rewrite the formula as:

If we divide both sides by the diameter, on the left side we will have , which is the ratio of circumference to diameter:

So, for any circle, the ratio of its circumference to its diameter is equal to , which is actually the definition of this very important mathematical constant.

### Example Question #2 : Calculating The Ratio Of Diameter And Circumference

For any given circle, what is the ratio of its diameter to its circumference?

Explanation:

To find the ratio between the diameter and the circumference of a circle, we need to use the formula for the circumference of a circle:

We can see this formula is in terms of radius, so we need to express it in a way that the circumference is in terms of diameter. Using the knowledge that the radius is half of the diameter:

Now that we have a simple formula involving the circumference and diameter, we can see that we will have the ratio of diameter to circumference if we divide both sides by the circumference. We then divide both sides by  to isolate the ratio of diameter to circumference and find our solution:

### Example Question #3 : Calculating The Ratio Of Diameter And Circumference

The circle with center , is inscribed in the square . What is the ratio of the diameter to the circumference of the circle given that the square has an area of ?

Explanation:

To calculate the ratio of the diameter to the circumference of the square we should first get the diameter, which is the same as the length of a side of the sqaure. To do so we just need to take the square root of the area of the square, which is 4. Also we should remember that the circumference is given by , where  is the length of the radius.

Now we should notice that this formula can also be written .

The ratio we are looking for is . Therefore, this ratio will always be  and this is our final answer.

### Example Question #4 : Calculating The Ratio Of Diameter And Circumference

Equilateral triangle  is inscribed in a circle. The perimeter of the triangle is . What is the radius of the circle?

Explanation:

The perimeter of the circle allows us to find the side of equilateral triangle ABC;  or 2.

From there, we can also find the length of the height of equilateral triangle ABC with the formula , which turns out to be .

Since in an equilateral triangle, the center of gravity is at  from any vertex, it follows that the radius of the circle is

### Example Question #5 : Calculating The Ratio Of Diameter And Circumference

A given circle has a circumference of  and a radius of . What is the ratio of the circle's circumference to its diameter?

Explanation:

For a given circle of circumference  and radius .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as

Therefore, the ratio of the circumference to the diameter of this and all other circles is .

### Example Question #6 : Calculating The Ratio Of Diameter And Circumference

A given circle has a circumference of  and a radius of . What is the ratio of the circle's circumference to its diameter?

Explanation:

For a given circle of circumference  and radius .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as

Therefore, the ratio of the circumference to the diameter of this and all other circles is,

.

### Example Question #7 : Calculating The Ratio Of Diameter And Circumference

A given circle has a circumference of  and a radius of . What is the ratio of the circle's circumference to its diameter?

Explanation:

For a given circle of circumference  and radius .

Since the radius  of a circle is equal to half of the circle's diameter , we can then define  as,

Therefore, the ratio of the circumference to the diameter of this circles is,

.

### Example Question #8 : Calculating The Ratio Of Diameter And Circumference

Find the circumference of a circle with a diameter measuring

Explanation:

The circumference of a circle is given by

where

We are told the diameter so we just need to plug in our value into the equation:

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