GMAT Math : Circles

Example Questions

arc of a circle measures . Give the radius of this circle.

Explanation:

arc of a circle is  of the circle. Since the length of this arc is , the circumference is  this, or

The radius of a circle is its circumference divided by ; therefore, the radius is

The arc  of a circle measures . The chord of the arc, , has length . Give the length of the radius of the circle.

Explanation:

A circle can be divided into  congruent arcs that measure

.

If the  (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one  chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord , or .

Example Question #12 : Calculating The Length Of A Radius

If a monster truck's wheels have circumference of ,  what is the distance from the ground to the center of the wheel?

Explanation:

If a monster truck's wheels have circumference of ,  what is the distance from the ground to the center of the wheel?

This question is asking us to find the radius of a circle. the distance from the outside of the circle to the center is the radius. We are given the circumference, so use the following formula:

Then, plug in what we know and solve for r

Two circles in the same plane have the same center. The smaller circle has radius 10; the area of the region between the circles is . What is the radius of the larger circle?

Explanation:

The area of a circle with radius  is .

Let  be the radius of the larger circle. Its area is . The area of the smaller  circle is . Since the area of the region between the circles is , and is the difference of these areas, we have

The smaller circle has radius .

Example Question #311 : Gmat Quantitative Reasoning

A circle on the coordinate plane has equation

Which of the following represents its circumference?

Explanation:

The equation of a circle centered at the origin is

where  is the radius of the circle.

In this equation, , so ; this simplifies to

The circumference of a circle is , so substitute :

Example Question #2 : Calculating Circumference

A circle on the coordinate plane has equation

What is its circumference?

Explanation:

The standard form of the area of a circle with radius  and center  is

Once we get the equation in standard form, we can find radius , and multiply it by  to get the circumference.

Complete the squares:

so  can be rewritten as follows:

,

so

And

On average, Stephanie walks  feet every  seconds. If Stephanie walks at her usual pace, how long will it take her to walk around a circular track with a radius of  feet, in seconds?

None of the other answers are correct.

seconds

seconds

seconds

seconds

seconds

Explanation:

The length of the track equals the circumference of the circle.

Therefore,  .

Example Question #2 : Calculating Circumference

A circle on the coordinate plane has equation .

What is its circumference?

Explanation:

The equation of a circle centered at the origin is

,

where  is the radius of the circle.

In the equation given in the question stem, , so .

The circumference of a circle is , so substitute :

Example Question #5 : Calculating Circumference

Let be concentric circles. Circle has a radius of , and the shortest distance from the edge of circle to the edge of circle is . What is the circumference of circle ?

Explanation:

Since are concentric circles, they share a common center, like sections of a bulls-eye target. Since the radius of is less than half the distance from the edge of to the edge of , we must have circle is inside of circle . (It's helpful to draw a picture to see what's going on!)

Now we can find the radius of by adding and , which is And the equation for finding circumfrence is . Plugging in for gives .

Consider the Circle :

(Figure not drawn to scale.)

Suppose Circle  represents a circular pen for Frank's mules. How many meters of fencing does Frank need to build this pen?