Circles - GMAT Quantitative

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Question

How many degrees does the hour hand on a clock move between 3 PM and 7:30 PM?

Answer

An hour hand rotates 360 degrees for every 12 hours, so the hour hand moves $\frac{360^{\circ}}{12\ hours}=30^{\circ}/hr$ .

There are 4.5 hours between 3 PM and 7:30 PM, so the total degree measure is

$4.5\ hours*30^{\circ}/hr = 135^{\circ}$ .

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Question

If a sector covers of a circle, what is the angle of the sector?

Answer

One full rotation of a circle is $360^{\circ}$ , so if a sector covers of a circle, its angle will be of $360^{\circ}$ . This gives us:

$\angle=0.20(360^{\circ})=\frac{1}{5}(360^{\circ})=72^{\circ}$

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Question

A given sector covers of a circle. What is the corresponding angle of the sector?

Answer

A circle comprises $360^\circ$ , so a sector comprising of the circle will have an angle that is of .

Therefore:

$\angle=0.70(360)=\frac{7}{10}(360)=252^{\circ}$

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Question

A given sector of a circle comprises of the circle. What is the corresponding angle of the sector?

Answer

A circle comprises $360^\circ$ , so a sector comprising of the circle will have an angle that is of .

Therefore:

$\angle=0.45(360)=\frac{9}{20}(360)=162^{\circ}$

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Question

The hour hand on a clock moves from 3PM to 6PM. How many degrees does the hour hand move?

Answer

The hour hand moves around a circle from 3PM to 6PM. Since there are 12 hours on a clock and the hand is moving through 3 of them, the hand is moving through a sector comprising of the circle because,

$\frac{3}{12}=\frac{1}{4}=0.25$ .

Since a circle has $360^\circ$ , the angle of the sector is:

$\angle=0.25(360^{\circ })=\frac{1}{4}(360^{\circ })=90^{\circ }$

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Question

The town of Thomasville organized a search party to look for a missing chicken. The party consisted of groups of people choosing a sector and searching outward from the center of town. Find the angle for the sector of searched by each group if each group chose an equal sized sector, and there were 120 groups.

Answer

The town of Thomasville organized a search party to look for a missing chicken. The party consisted of groups of people choosing a sector and searching outward from the center of town. Find the angle for the sector of searched by each group if each group chose an equal sized sector, and there were 120 groups.

Begin by dissecting the question and figurign out exactly what they are asking and telling you. It's a bit wordy, but what we are looking for is the measure of the central angle for each of the search-sectors

We are told that there are 120 equal sectors.

We also know that a circle is made up of $360^{\circ}$

So, to find the central angle of each sector, simply do the following calculation:

$\frac{360^{\circ}}{120}=3^{\circ}/sector$

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Question

Note: Figure NOT drawn to scale.

$m \angle A = 65 ^{\circ }; m \angle B = 55 ^{\circ }$ .

Order the degree measures of the arcs $\widehat{AB}, \widehat{BC}, \widehat{CA}$ from least to greatest.

Answer

$m \angle C = 180 ^{\circ } - m \angle A - m \angle B = 180 ^{\circ } - 65^{\circ } - 55^{\circ } = 60^{\circ }$

$m \angle B < m \angle C < m \angle A$ , so, by the Multiplication Property of Inequality,

$2 \cdot m \angle B < 2 \cdot m \angle C < 2 \cdot m \angle A$ .

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

$m \widehat{ AC} < m\widehat{ AB } < m\widehat{ BC}$ .

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Question

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.

Find the value of central angle .

Answer

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 .

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

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 :

$\frac{\Theta }{360}=\frac{1}{6}$

$\Theta=\frac{1}{6}*360^{\circ}=60^{\circ}$

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

$\angle AOC= 360^{\circ}-\angle AOB - \angle BOC= 360^{\circ}-180^{\circ}-60^{\circ}=120^{\circ}$

So, $\angle AOC=120^{\circ}$ .

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Question

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?

Answer

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 $A_{a}= \pi \left ( 4r\right )^{2} = 16 \pi r^{2}$ ; that of the square is $A_{b}= r^{2}$ . Therefore, a sector of the circle with area $r^{2}$ will be $\frac{1}{16 \pi}$ of the circle, which is a sector of measure

$\frac{1}{16 \pi} \times 360 ^{\circ } = \left (\frac{45}{2\pi} \right )^{\circ }$

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Question

Angle $\widehat{DCB}$ is $35^{\circ}$ . What is angle $\widehat{DAB}$ ?

Answer

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, $\widehat{DAB}$ is twice $\widehat{DCB}$ .

Thus,

$2(35^\circ)=70^\circ$

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Question

are evenly spaced points on the circle. What is angle $\widehat{DFC}$ ?

Answer

We can see that the points devide the $360^{\circ}$ of the circle in 5 equal portions.

The final answer is given simply by $\frac{360}{5}$ which is $72^{\circ}$ , this is the angle of a slice of a pizza cut in 5 parts if you will!

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Question

The points and are evenly spaced on the circle of center . What is the size of angle $\widehat{FBE}$ ?

Answer

As we have seen previously, the 6 points divide the $360^{\circ}$ of the circle in 6 portion of same angle. Each portion form an angle of $\frac{360}{6}$ 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, $\widehat{FGE}=2\widehat{FBE}$ . Since $\widehat{FGE}$ is 60 degrees, than, $\widehat{FBE}$ must be 30 degrees, this is our final answer.

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Question

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?

Answer

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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Question

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.

Find the area of sector .

Answer

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 . Thus, our total area is:

$A=16 \pi :cm^2$

We need to go one step further to find the area of sector . Simply multiply the total area by the fractional amount that sector 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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Question

Consider the Circle :

(Figure not drawn to scale.)

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

Answer

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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Question

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

Answer

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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Question

For $10, Brandon can order either a 12"-diameter pizza, two 6"-diameter pizzas, or three 4"-diameter pizzas. Which option is the best value, assuming all pizzas are the same thickness?

Answer

For a circle, $Area = \pi r ^{2}$ .

Therefore, the area of the 12" pizza $=6^{2}\pi =36\pi$ .

The area of the two 6" pizzas $=2\times 3^{2}\pi=2\times 9\pi =18\pi$ .

The area of the three 4" pizzas = $3\times 2^{2}\pi=3\times 4\pi =12\pi$ .

The 12" pizza is the best option.

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Question

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 $60 \pi$ .

Answer

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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Question

Note: Figure NOT drawn to scale

Refer to the above diagram.

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

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

What is $m \angle BAC$ ?

Answer

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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Question

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?

Answer

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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