GMAT Math : Circles

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Example Question #5 : Calculating The Percentage Of A Sector From An Angle

What percentage of a circle is a sector if the angle of the sector is $27^{\circ}$ ?

Possible Answers:

$7.5\%$

$10\%$

$5\%$

$27\%$

$13.5\%$

Correct answer:

$7.5\%$

Explanation:

The full measure of a circle is $360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ . We are given a sector with an angle of $27^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $27^{\circ}$ is of $360^{\circ}$ . This gives us:

$\frac{27^{\circ}}{360^{\circ}}=\frac{3}{40}=0.075=7.5\%$

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Example Question #6 : Calculating The Percentage Of A Sector From An Angle

What percentage of a circle's total area is covered by a sector with an angle of $54^{\circ}$ ?

Possible Answers:

$27\%$

$15\%$

$54\%$

$10\%$

$12.5\%$

Correct answer:

$15\%$

Explanation:

The full measure of a circle is $360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ . We are given a sector with an angle of $54^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $54^{\circ}$ is of $360^{\circ}$ . This gives us:

$\frac{54^{\circ}}{360^{\circ}}=\frac{3}{20}=0.15=15\%$

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Example Question #1 : Calculating The Area Of A Sector

A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36 degree sector for himself, what is the area of his slice of pizza, rounded to the nearest square inch?

Possible Answers:

Correct answer:

Explanation:

First we calculate the area of the pizza. The area of a circle is defined as $\pi r^2$ . Since our diameter is 18 inches, our radius is 18/2 = 9 inches. So the total area of the pizza is $\pi 9^2 = 81\pi = 254.469$ square inches.

Since the sector of the pie he cut for himself is 36 degrees, we can set up a ratio to find how much of the pizza he cut for himself. Let x be the area of the pizza he cut for himself. Then we know, $\frac{36^\circ}{360^\circ} = \frac{x}{254.469}$

Solving for x, we get x=25.45 square inches, which rounds down to 25.

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Example Question #2 : Calculating The Area Of A Sector

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

Find the area of sector COB .

Possible Answers:

$\frac{4\pi}{3} \:cm^2$

$\frac{8\pi}{3} \:cm^2$

$\frac{8\pi}{5} \:cm^2$

$18\pi \:cm^2$

$\frac{1\pi}{6} \:cm^2$

Correct answer:

$\frac{8\pi}{3} \:cm^2$

Explanation:

To find the area of a sector, we need to know the total area as well as the fractional amount of the sector at which we are looking.

In this case, we find the total area by using the following equation:

$A=\pi r^2$

Because line segment is our diameter, our radius is $4\:cm$ . Thus, our total area is:

$A=16 \pi \:cm^2$

We need to go one step further to find the area of sector COB . Simply multiply the total area by the fractional amount that sector COB covers. We are told it is $\frac{1}{6}th$ of the circle's area, so do the following:

$A_{COB}=16\pi*\frac{1}{6}=\frac{16\pi}{6}=\frac{8\pi}{3} \:cm^2$

Thus, our answer is $\frac{8\pi}{3} \:cm^2$ .

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Example Question #3 : Calculating The Area Of A Sector

Consider the Circle :

Circle3

(Figure not drawn to scale.)

If angle $\angle BOX$ is $120^{\circ}$ , what is the area of sector BOX in square meters?

Possible Answers:

$75\:m^2$

$225 \pi\:m^2$

$15 \pi\:m^2$

$115 \pi\:m^2$

$75 \pi\:m^2$

Correct answer:

$75 \pi\:m^2$

Explanation:

To find the area of a sector, simply multiply the total area of the circle by the fraction of the part you are looking at.

In this case, our area will come from the following:

$A=\pi r^2=\pi*(15\:m)^2=225 \pi\:m^2$

To find the fractional part of the circle we care about, take the number of degrees in $\angle BOX$ over the total number of degrees in a circle ( $360^{\circ}$ ):

$\frac{120}{360}=\frac{1}{3}$

So, we find our answer by multiplying these two parts together:

$225 \pi\:m^2 * \frac{1}{3}=75 \pi\:m^2$

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Example Question #1 : Calculating The Area Of A Sector

If a circle has an area of $120\pi$ , what is the area of a sector with an angle of $36^{\circ}$ ?

Possible Answers:

$6\pi$

$72\pi$

$60\pi$

$24\pi$

$12\pi$

Correct answer:

$12\pi$

Explanation:

The area of a sector with a certain angle will be whatever fraction of the total circle's area the angle of the sector is of $360^{\circ}$ . This means we divide $36^{\circ}$ by $360^{\circ}$ , and then multiply that fraction by the total area of the circle to give us the area of the sector:

$A=\frac{36^{\circ}}{360^{\circ}}(120\pi )=\frac{1}{10}120\pi=12\pi$

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Example Question #5 : Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: Arc $\widehat{AB}$ has length $30 \pi$ .

Statement 2: Arc $\widehat{BD}$ has length $60 \pi$ .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{AB}$ is, and therefore, the central angle of the sector cannot be determined. Also, no information about the circle can be determined. A similar argument can be given for Statement 2 being insufficient.

Now assume both statements are true. Then the length of semicircle $\widehat{ABD}$ is equal to $30 \pi + 60 \pi = 90 \pi$ . The circumference is twice this, or $180 \pi$ . The radius can be calculated as $r = C \div( 2 \pi )=180 \pi \div (2 \pi) = 90$ , and the area, $\pi r^{2}= \pi \cdot 90^{2} = 8,100 \pi$ . Also, $\widehat{AB}$ is $\frac{30 \pi}{180 \pi}= \frac{1}{6}$ of the circle, and the area of the sector can now be calculated as $\frac{1}{6} \times 8,100 \pi = 1,350 \pi$ .

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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

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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GMAT Math : Circles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Calculating The Percentage Of A Sector From An Angle

Example Question #6 : Calculating The Percentage Of A Sector From An Angle

Example Question #1 : Calculating The Area Of A Sector

Example Question #2 : Calculating The Area Of A Sector

Example Question #3 : Calculating The Area Of A Sector

Example Question #1 : Calculating The Area Of A Sector

Example Question #5 : Calculating The Area Of A Sector

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

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