# GMAT Math : Circles

## Example Questions

### Example Question #21 : Calculating The Area Of A Circle

What is the area of a circle with a diameter of ?

Explanation:

The area  of a circle is defined by , where  is the radius of the circle. We are provided with the diameter  of the circle, which is twice the length of .

If , then

Then, solving for :

### Example Question #21 : Radius

A circle on the coordinate plane is defined by the equation . What is the area of the circle?

Not enough information provided.

Explanation:

The equation of a circle centered at the origin of the coordinate plane is , where  is the radius of the circle.

The area  of the circle, in turn, is defined by the equation .

Since we are provided with the equation , we can deduce that  and that .

### Example Question #22 : Calculating The Area Of A Circle

What is the area of a circle with a diameter of ?

Not enough information provided.

Explanation:

The area  of a circle is defined by , where  is the radius of the circle. We are provided with the diameter  of the circle, which is twice the length of .

If , then

Therefore:

### Example Question #22 : Calculating The Area Of A Circle

A square is inscribed in a circle of

What is the area of the region inside the circle but not inside the square?

Explanation:

Our first step will be to draw two radii, from the circle's center to two adjacent vertexes of the square, forming a triangle, the third side of which is the edge of the square. Because these radii bisect the square's right angles, we can determine that the triangle we drew is a 45˚ - 45˚ - 90˚ right triangle. Because of the rules governing those isoceles right triangles, we can then determine that, because the two legs of the triangle are 5 (radius = 5), the hypotenuse (the square's side) must equal .

Now we can find the square's area:

and the circle's area:

Finally we subtract the square's area from the circle's area:

### Example Question #21 : Calculating The Area Of A Circle

An engineer is designing a circular hatch for a submarine. If the hatch must have a circumference of , what will its area be?

Explanation:

An engineer is designing a circular hatch for a submarine. If the hatch must have a circumference of , what will its area be?

To find area, we will need the radius. We can find the radius using the following formula:

So, plug in and solve for r

Next, use the area formula to find the area:

### Example Question #56 : Circles

and  are the area and the diameter of the same circle.

.

Which of the following is a true statement?

varies directly as the fourth power of .

varies inversely as the fourth power of .

varies directly as the square of .

varies inversely as the square of .

varies directly as the square root of .

varies inversely as the fourth power of .

Explanation:

The area and the diameter of a circle are related by the formula

Substituting:

If , then

,

and  varies inversely as the fourth power of .

### Example Question #291 : Gmat Quantitative Reasoning

A circle has radius . Give its area.

Explanation:

The area of a circle is found using the following formula:

Set :

### Example Question #21 : Radius

If the pitcher plant Sarracenia purpurea has a circular opening with a circumference of 6 inches, what is the area of the opening?

Explanation:

If the pitcher plant Sarracenia purpurea has a circular opening with a circumference of 6 inches, what is the area of the opening?

We need to work backward from circumference to find area.

Circumference can be found as follows:

Use this to find "r" which we will use to find the area:

Next, find area using the following:

### Example Question #51 : Circles

A square and a circle have the same area. What is the ratio of the length of one side of the square to the radius of the circle?

Explanation:

Let  be the sidelength of the square is the square and  be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

We find the ratio of  to  - that is,  - as follows:

The correct ratio is .

### Example Question #291 : Gmat Quantitative Reasoning

The radius of Circle A is three times the radius of Circle B, which is, in turn, equal to the diameter of Circle C. The sum of their circumferences is . What is the radius of Circle C?

Explanation:

Let  be the radius of circle C. Then Circle B has radius equal to twice this, or , and Circle A has radius equal to three times that, or . The circumference of a circle is equal to  times the radius, so the circumferences of the three circles are:

A:

B:

C:

These circumferences add up to , so set up and solve the equation: