GMAT Math : Calculating the length of an arc

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Example Question #1 : Calculating The Length Of An Arc

Chords

Note: Figure NOT drawn to scale

Refer to the above diagram.

$\widehat{ABC} = 220 ^{\circ }$

$\widehat{BCA} = 230 ^{\circ }$

What is $m \angle BAC$ ?

Possible Answers:

$55^{\circ }$

$40^{\circ }$

$60^{\circ }$

$50^{\circ }$

$45^{\circ }$

Correct answer:

$45^{\circ }$

Explanation:

The degree measure of $\angle BAC$ is half the degree measure of the arc it intercepts, which is $\widehat{BC}$ . We can use the measures of the two given major arcs to find $m \widehat{BC}$ , then take half of this:

$m\widehat{AC} = 360 ^{\circ } - m\widehat{ABC} = 360 ^{\circ } - 220 ^{\circ } = 140^{\circ }$

$m\widehat{AB} = 360 ^{\circ } - m\widehat{BCA} = 360 ^{\circ } - 230 ^{\circ } = 130^{\circ }$

$m\widehat{BC} +m\widehat{AB} + m\widehat {AC} =360 ^{\circ }$

$m\widehat{BC} +130^{\circ }+ 140^{\circ } =360 ^{\circ }$

$m\widehat{BC} +270^{\circ } =360 ^{\circ }$

$m\widehat{BC} +270^{\circ } -270^{\circ } =360 ^{\circ }-270^{\circ }$

$m\widehat{BC} =90 ^{\circ }$

$\angle BAC = \frac{1}{2}m\widehat{BC} = \frac{1}{2} \cdot 90 ^{\circ } = 45^{\circ }$

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Example Question #1 : 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?

Possible Answers:

$448 \pi \textrm{ in}$

$128 \pi \textrm{ in}$

$512 \pi \textrm{ in}$

$256 \pi \textrm{ in}$

$112 \pi \textrm{ in}$

Correct answer:

$512 \pi \textrm{ in}$

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 $2 \frac{2}{3} = \frac{8}{3}$ revolutions.

In one revolution, the tip of an eight-foot minute hand moves $C = 2 \pi r =2 \pi \cdot 8 = 16 \pi$ feet, or $16 \pi \cdot 12 = 192 \pi$ inches.

After $\frac{8}{3}$ revolutions, the tip of the minute hand has moved $192 \pi \cdot 2 \frac{2}{3} = \frac{192}{1} \cdot \frac{8}{3} \cdot \pi = 512 \pi$ inches.

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Example Question #3 : 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 $8\:cm$ . Points , , and are on the circle. Sector COB covers $\frac{1}{6}th$ of the total area of the circle. Answer the following questions regarding this shape.

Circle1

What is the length of the arc formed by angle AOB ?

Possible Answers:

$4 \pi \:cm$

$12\pi \:cm$

$8 \pi \:cm$

$16 \pi \:cm$

$64 \pi \:cm$

Correct answer:

$4 \pi \:cm$

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:

$C=2\pi *r=d \pi$

Where is our radius and is our diameter.

In this problem, our diameter is the length of , which is $8\:cm$ , so our total circumference is:

$C=8\pi \:cm$

Now, to find the fraction of the circle we are interested in, we need to realize that angle AOB is 180 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:

$Arc Length=d \pi *\frac{\Theta }{360}=8 \pi *\frac{180 }{360}=8 \pi *\frac{1 }{2}=4 \pi \:cm$

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Example Question #4 : Calculating The Length Of An Arc

Consider the Circle :

Circle3

(Figure not drawn to scale.)

Suppose $\angle BOX$ is $100^{\circ}$ . What is the measure of arc in meters?

Possible Answers:

$\frac{5}{3}\pi\:m$

$18\pi\:m$

$5\pi\:m$

$\frac{25}{3}\pi\:m$

$30\pi\:m$

Correct answer:

$\frac{25}{3}\pi\:m$

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:

$C=2 \pi r=2*(15\:m)*\pi=30 \pi\:m$

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

$\frac{100}{360}=\frac{10}{36}=\frac{5}{18}$

Multiply together and simplify:

$Arc\:Length=\frac{5}{18}*30 \pi= \frac{150\pi}{18}= \frac{25}{3}\pi$

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Example Question #5 : Calculating The Length Of An Arc

What is the arc length for a sector with a central angle of $45^{\circ}$ if the radius of the circle is ?

Possible Answers:

$\frac{3\pi }{2}$

$\frac{\pi }{2}$

$\frac{\pi }{4}$

$\pi$

$\frac{3\pi }{4}$

Correct answer:

$\frac{3\pi }{4}$

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 $\frac{\pi }{180}$ :

$S=r\theta$

$S=3(45^{\circ})(\frac{\pi }{180^{\circ}})=3(\frac{\pi }{4})=\frac{3\pi}{4}$

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Example Question #6 : Calculating The Length Of An Arc

The arc $\widehat{AB}$ of a circle measures $90^{\circ }$ . The chord of the arc, $\overline{AB}$ , has length x + 2 . Give the length of the arc $\widehat{AB}$ .

Possible Answers:

$\frac{2 \sqrt{2}}{\pi}x + \frac{4 \sqrt{2}}{\pi}$

$\frac{ \sqrt{2}}{\pi}x + \frac{ \sqrt{2}}{\pi}$

$\frac{ \pi \sqrt{2}}{2} x + \pi \sqrt{2}$

$\frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

$\pi x \sqrt{2}+ 2 \pi \sqrt{2}$

Correct answer:

$\frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

Explanation:

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

Circle x

If we extend the figure to depict the circle as the composite of four quarter-circles, each a $90^{\circ }$ arc, we see that x+2 is also the side of an inscribed square. A diagonal of this square, which measures $\sqrt{2}$ times this sidelength, or

$(x+2) \sqrt{2} = x\sqrt{2} + 2\sqrt{2}$ ,

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

$\pi ( x\sqrt{2} + 2\sqrt{2}) = x \pi \sqrt{2} + 2 \pi \sqrt{2}$ .

Since a $90^{\circ }$ arc is one fourth of a circle, the length of arc $\widehat{AB}$ is

$\frac{1}{4}( x \pi \sqrt{2} + 2 \pi \sqrt{2})$

$= \frac{ \pi }{4} x\sqrt{2} + \frac{ \pi }{2} \sqrt{2}$

$= \frac{ \pi \sqrt{2}}{4} x + \frac{ \pi \sqrt{2}}{2}$

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GMAT Math : Calculating the length of an arc

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Example Question #1 : Calculating The Length Of An Arc

Example Question #1 : Calculating The Length Of An Arc

Example Question #3 : Calculating The Length Of An Arc

Example Question #4 : Calculating The Length Of An Arc

Example Question #5 : Calculating The Length Of An Arc

Example Question #6 : Calculating The Length Of An Arc

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