Calculating the length of an arc - GMAT Quantitative

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Note: Figure NOT drawn to scale

Refer to the above diagram.

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

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

What is $m \angle BAC$ ?

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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 }$$

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?

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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.

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 $$\frac{1}{6}$th$ of the total area of the circle. Answer the following questions regarding this shape.

What is the length of the arc formed by angle ?

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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 , 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 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:

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

Consider the Circle :

(Figure not drawn to scale.)

Suppose $\angle BOX$ is . What is the measure of arc in meters?

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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$

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

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

$S=r\theta$

The arc $\widehat{AB}$ of a circle measures . The chord of the arc, $\overline{AB}$ , has length . Give the length of the arc $\widehat{AB}$ .

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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 $\sqrt{2}$ 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 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}$

Note: Figure NOT drawn to scale

Refer to the above diagram.

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

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

What is $m \angle BAC$ ?

Tap to see back →

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 }$$

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?

Tap to see back →

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.

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 $$\frac{1}{6}$th$ of the total area of the circle. Answer the following questions regarding this shape.

What is the length of the arc formed by angle ?

Tap to see back →

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 , 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 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:

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

Consider the Circle :

(Figure not drawn to scale.)

Suppose $\angle BOX$ is . What is the measure of arc in meters?

Tap to see back →

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$

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

Tap to see back →

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

$S=r\theta$

The arc $\widehat{AB}$ of a circle measures . The chord of the arc, $\overline{AB}$ , has length . Give the length of the arc $\widehat{AB}$ .

Tap to see back →

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 $\sqrt{2}$ 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 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}$

Note: Figure NOT drawn to scale

Refer to the above diagram.

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

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

What is $m \angle BAC$ ?

Tap to see back →

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 }$$

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?

Tap to see back →

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.

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 $$\frac{1}{6}$th$ of the total area of the circle. Answer the following questions regarding this shape.

What is the length of the arc formed by angle ?

Tap to see back →

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 , 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 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:

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

Consider the Circle :

(Figure not drawn to scale.)

Suppose $\angle BOX$ is . What is the measure of arc in meters?

Tap to see back →

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$

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

Tap to see back →

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

$S=r\theta$

The arc $\widehat{AB}$ of a circle measures . The chord of the arc, $\overline{AB}$ , has length . Give the length of the arc $\widehat{AB}$ .

Tap to see back →

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 $\sqrt{2}$ 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 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}$

Note: Figure NOT drawn to scale

Refer to the above diagram.

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

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

What is $m \angle BAC$ ?

Tap to see back →

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 }$$

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?

Tap to see back →

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.