# GMAT Math : Circles

## Example Questions

### Example Question #21 : Sectors

Consider the Circle :

(Figure not drawn to scale.)

Suppose  is . What is the measure of arc  in meters?

Explanation:

To find arc length, multiply the total circumference of a circle by the the fraction of the total circle that defines the arc with the length for which you are solving.

In this case, to find the total circumference:

To find the fraction with which we are concerned, make a fraction with the number of degrees in  in the numerator and the total degrees in a circle in the denominator:

Multiply together and simplify:

### Example Question #5 : Calculating The Length Of An Arc

What is the arc length for a sector with a central angle of  if the radius of the circle is  ?

Explanation:

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by  :

### Example Question #6 : Calculating The Length Of An Arc

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

Explanation:

Examine the figure below, which shows the arc and chord in question.

If we extend the figure to depict the circle as the composite of four quarter-circles, each a  arc, we see that  is also the side of an inscribed square. A diagonal of this square, which measures  times this sidelength, or

,

is a diameter of this circle. The circumference is  times the diameter, or

.

Since a  arc is one fourth of a circle, the length of arc  is

### Example Question #21 : Geometry

Note: Figure NOT drawn to scale.

.

Order the degree measures of the arcs  from least to greatest.

Explanation:

, so, by the Multiplication Property of Inequality,

.

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as

.

### Example Question #2 : Calculating The Angle Of A Sector

In the figure shown below, line segment  passes through the center of the circle and has a length of . Points , and  are on the circle. Sector  covers  of the total area of the circle. Answer the following questions regarding this shape.

Find the value of central angle .

Explanation:

Here we need to recall the total degree measure of a circle. A circle always has exactly  degrees.

Knowing this, we need to utilize two other clues to find the degree measure of .

1) Angle  measures  degrees, because it is made up of line segment , which is a straight line.

2) Angle  can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle :

So, to find angle , we just need to subtract our other values from :

So, .

### Example Question #22 : Geometry

The radius of Circle A is equal to the perimeter of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

Explanation:

Call the length of a side of Square B . Its perimeter is , which is the radius of Circle A.

The area of the circle is ; that of the square is . Therefore, a sector of the circle with area  will be  of the circle, which is a sector of measure

### Example Question #21 : Circles

Angle  is . What is angle  ?

Explanation:

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words,  is twice .

Thus,

### Example Question #21 : Sectors

are  evenly spaced points on the circle. What is angle ?

Explanation:

We can see that the points devide the  of the circle in 5 equal portions.

The final answer is given simply by  which is , this is the angle of a slice of a pizza cut in 5 parts if you will!

### Example Question #261 : Gmat Quantitative Reasoning

The  points  and  are evenly spaced on the circle of center . What is the size of angle ?

Explanation:

As we have seen previously, the 6 points divide the  of the circle in 6 portion of same angle. Each portion form an angle of  or 60 degrees. As we also have previously seen, the angle formed by the lines intercepting an arc is twice more at the center of the circle than at the intersection of the lines intercepting the same arc with the circle, provided one of these lines is the diameter. In other words, . Since  is 60 degrees, than,  must be 30 degrees, this is our final answer.

### Example Question #1 : Calculating The Area Of A Circle

A circle is inscribed in a square with area 100.  What is the area of the circle?