GMAT Math : Calculating the length of an arc

Example Questions

Example Question #17 : Circles

Note: Figure NOT drawn to scale

Refer to the above diagram.

What is  ?

Explanation:

The degree measure of  is half the degree measure of the arc it intercepts, which is . We can use the measures of the two given major arcs to find , then take half of this:

Example Question #2 : Calculating The Length Of An Arc

A giant clock has a minute hand that is eight feet long. The time is now 2:40 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Explanation:

Between noon and 2:40 PM, two hours and forty minutes have elapsed, or, equivalently, two and two-thirds hours. This means that the minute hand has made   revolutions.

In one revolution, the tip of an eight-foot minute hand moves  feet, or  inches.

After   revolutions, the tip of the minute hand has moved  inches.

Example Question #1 : Calculating The Length Of An Arc

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.

What is the length of the arc formed by angle ?

Explanation:

To find arc length, we need to find the total circumference of the circle and then the fraction of the circle we are interested in. Our circumference of a circle formula is:

Where  is our radius and  is our diameter.

In this problem, our diameter is the length of , which is , so our total circumference is:

Now, to find the fraction of the circle we are interested in, we need to realize that angle  is  degrees. We know this because it is made by straight line . Armed with this knowledge, we can safely calculate the length of our arc using the following formula:

Example Question #21 : Circles

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 #22 : Circles

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 #23 : Circles

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