Circles - GMAT Quantitative

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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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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.

What is the length of the arc formed by angle ?

Answer

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:

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

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Question

Consider the Circle :

(Figure not drawn to scale.)

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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 $90^{\circ }$ 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 $90^{\circ }$ 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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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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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.

What is the length of the arc formed by angle ?

Answer

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:

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

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Question

Consider the Circle :

(Figure not drawn to scale.)

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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 $90^{\circ }$ 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 $90^{\circ }$ 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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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

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

For any circle, what is the ratio of its circumference to its diameter?

Answer

In order to calculate the ratio of circumference to diameter, we need an equation that involves both variables. The formula for circumference is as follows:

$C=2\pi r$

We need to express the radius in terms of diameter. The radius of a circle is half of its diameter, so we can rewrite the formula as:

$C=2\pi r=2\pi (\frac{d}{2})=\pi d$

If we divide both sides by the diameter, on the left side we will have $\frac{C}{d}$ , which is the ratio of circumference to diameter:

$C=\pi d$

$\frac{C}{d}=\pi$

So, for any circle, the ratio of its circumference to its diameter is equal to $\pi$ , which is actually the definition of this very important mathematical constant.

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Question

For any given circle, what is the ratio of its diameter to its circumference?

Answer

To find the ratio between the diameter and the circumference of a circle, we need to use the formula for the circumference of a circle:

$C=2\pi r$

We can see this formula is in terms of radius, so we need to express it in a way that the circumference is in terms of diameter. Using the knowledge that the radius is half of the diameter:

$C=2\pi r\rightarrow C=2\pi (\frac{d}{2})\rightarrow C=\pi d$

Now that we have a simple formula involving the circumference and diameter, we can see that we will have the ratio of diameter to circumference if we divide both sides by the circumference. We then divide both sides by $\pi$ to isolate the ratio of diameter to circumference and find our solution:

$C=\pi d$

$\frac{d}{C}=\frac{1}{\pi }$

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Question

The circle with center , is inscribed in the square . What is the ratio of the diameter to the circumference of the circle given that the square has an area of ?

Answer

To calculate the ratio of the diameter to the circumference of the square we should first get the diameter, which is the same as the length of a side of the sqaure. To do so we just need to take the square root of the area of the square, which is 4. Also we should remember that the circumference is given by $2\pi r$ , where is the length of the radius.

Now we should notice that this formula can also be written $d\pi$ .

The ratio we are looking for is $\frac{d}{d\pi}=\frac{1}{\pi}$ . Therefore, this ratio will always be $\frac{4}{4\pi}=\frac{1}{\pi}$ and this is our final answer.

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