Circles - ISEE Upper Level Quantitative Reasoning

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Question

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

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Answer

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

Begin by dividing over the 100

Then multiply by 360

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Question

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

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Answer

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

To find an angle measure from a percentage, simply convert the percentage to a decimal and then multiply it by 360 degrees.

$45%\rightarrow 0.45$

$0.45 * $360^{\circ}$$=162^{\circ}$$

So, our answer is 162 degrees.

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Question

Sector SOW has a central angle of . What percentage of the circle does it cover?

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Answer

Sector SOW has a central angle of . What percentage of the circle does it cover?

Recall that there is a total of 360 degrees in a circle. SOW occupies 45 of them. To find the percentage, simply do the following:

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Question

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . What percentage of the circle is highlighted?

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Answer

While visiting a history museum, you see a radar display which consists of a circular screen with a highlighted wedge with an angle of . What percentage of the circle is highlighted?

To find the percentage of a sector, simply put the degree measure of the angle over 360 and multiply by 100.

So, our answer is 18.06%

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Question

What is the radius of a circle with circumference equal to $36\pi$ ?

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Answer

The circumference of a circle can be found using the following equation:

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

$36\pi=2r\pi$

$\frac{36\pi}{\pi}$=\frac{2r\pi}{\pi}$

$\frac{36}{2}$=\frac{2r}{2}$

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Question

What is the value of the radius of a circle if the area is equal to $18\pi$ ?

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Answer

The equation for finding the area of a circle is $\pi $r^{2}$$ .

Therefore, the equation for finding the value of the radius in the circle with an area of $18\pi$ is:

$\pi $r^{2}$=18\pi$

$r=$\sqrt{18}$=$\sqrt{9\cdot 2}$=3$\sqrt{2}$$

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Question

What is the radius of a circle with a circumference of $24\pi$ ?

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Answer

The circumference of a circle can be found using the following equation:

$C=2r\pi$

We plug in the circumference given, $24\pi$ into and use algebraic operations to solve for .

$24\pi=2r\pi$

$\frac{24\pi }{\pi }$=\frac{2r\pi }{\pi }$

$\frac{24}{2}$=\frac{2r}{2}$

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Question

Refer to the above diagram. $\overarc {ACB}$ has length $60 \pi$ . Give the radius of the circle.

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Answer

Inscribed $\angle ACB$ , which measures $72 ^{\circ }$ , intercepts a minor arc with twice its measure. That arc is $\overarc {AB}$ , which consequently has measure

$72 ^{\circ } \times 2 = 144 ^{\circ }$ .

The corresponding major arc, $\overarc {ACB}$ , has as its measure

$360 ^{\circ } - 144 ^{\circ } = 216 ^{\circ }$ , and is

$\frac{ 216 }{360 }$ = $\frac{ 216 \div 72 }{360 \div 72 }$ = $\frac{3}{5}$

of the circle.

If we let be the circumference and be the radius, then $\overarc {ACB}$ has length

$$\frac{3}{5}$ C = $\frac{3}{5}$ \cdot 2 \pi r = $\frac{6 \pi}{5}$ r$ .

This is equal to $60 \pi$ , so we can solve for in the equation

$$\frac{6}{5}$ \pi r = 60 \pi$

$$\frac{5}{6 \pi }$ \cdot $\frac{6 \pi }{5}$ r = $\frac{5}{6 \pi }$ \cdot 60 \pi$

The radius of the circle is 50.

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Question

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

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Answer

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

Begin with the formula for circumference of a circle:

$Circumference=2\pi r$

Now, plug in our known and work backwards:

$768\pi m=2\pi r$

Divide both sides by two pi to get:

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Question

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi $m^2$$ .

What is the radius of the crater?

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Answer

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi $m^2$$ .

What is the radius of the crater?

To solve this, we need to recall the formula for the area of a circle.

$A=\pi $r^2$$

Now, we know A, so we just need to plug in and solve for r!

$169 \pi $m^2$=\pi r ^2$

Begin by dividing out the pi

Then, square root both sides.

So our answer is 13m.

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Question

What is the area of a circle with a diameter of , rounded to the nearest whole number?

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Answer

The formula for the area of a circle is

pi $r^{2}$

Find the radius by dividing 9 by 2:

$\frac{9}{2}$=4.5

So the formula for area would now be:

pi $r^{2}$=pi $(4.5)^{2}$=20.25pi approx 63.6= 64

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Question

What is the area of a circle that has a diameter of inches?

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Answer

The formula for finding the area of a circle is $\pi $r^{2}$$ . In this formula, represents the radius of the circle. Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius. In order to do this, we divide the diameter by .

$$\frac{15}{2}$=7.5$

Now we use for in our equation.

$\pi $(7.5)^{2}$=176.7146 : $in^{2}$$

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Question

What is the area of a circle with a diameter equal to 6?

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Answer

First, solve for radius:

$r=\frac{d}{2}$=\frac{6}{2}$=3$

Then, solve for area:

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Question

The diameter of a circle is $4\ cm$ . Give the area of the circle.

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Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi $d^2$}{4}$$ ,

where is the diameter of the circle, and $\pi$ is approximately .

$Area=\frac{\pi $d^2$}{4}$=\frac{\pi\times $4^2$}{4}$=4\pi \Rightarrow Area\approx 4\times 3.14\Rightarrow Area\approx 12.56 \ $cm^2$$

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Question

The diameter of a circle is . Give the area of the circle in terms of .

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Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi $d^2$}{4}$$ ,

where is the diameter of the circle and $\pi$ is approximately .

$Area=\frac{\pi $(4t)^2$}{4}$=\frac{16\pi $t^2$}{4}$=4\pi $t^2$ \Rightarrow Area\approx 4\times 3.14\times $t^2$$

$\Rightarrow Area\approx $12.56t^2$$

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Question

The radius of a circle is $$\frac{2}{\sqrt{\pi }$}$ . Give the area of the circle.

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Answer

The area of a circle can be calculated as $Area=\pi $r^2$$ , where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi $r^2$=\pi\times $($\frac{2}{\sqrt{\pi}$})^2$=\pi\times $\frac{4}{\pi}$\Rightarrow Area=4$

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Question

The circumference of a circle is inches. Find the area of the circle.

Let $\pi = 3.14$ .

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Answer

First we need to find the radius of the circle. The circumference of a circle is $Circumference =2\pi r$ , where is the radius of the circle.

$12.56=2\times 3.14\times r\Rightarrow r=2\ in$

The area of a circle is $Area=\pi $r^2$$ where is the radius of the circle.

$Area=\pi $r^2$=3.14\times $2^2$=12.56\ $in^2$$

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Question

The perpendicular distance from the chord to the center of a circle is $$\sqrt{2}$t$ , and the chord length is $2$\sqrt{2}$t$ . Give the area of the circle in terms of .

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Answer

Chord length = , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length =

$\Rightarrow $8t^2$$=4(r^2$$-2t^2$)\Rightarrow $4r^2$$=16t^2$\Rightarrow $r^2$$=4t^2$\Rightarrow r=2t$

$Area=\pi $r^2$$ , where is the radius of the circle and $\pi$ is approximately .

$Area=\pi $r^2$=3.14\times $(2t)^2$$=12.56t^2$$

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Question

In the above figure, .

What percent of the figure is shaded gray?

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Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ :

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi $r^{2}$ = \pi \cdot $2^{2}$ =4 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

The outer gray ring is the region between the largest and second-largest circles, and has area

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$

The inner gray ring is the region between the second-smallest and smallest circles, and has area

$A _{2} - A _{1} = 4 \pi - \pi = 3 \pi$

The total area of the gray regions is

$7 \pi + 3 \pi = 10 \pi$

Since $10 \pi$ out of total area $16 \pi$ is gray, the percent of the figure that is gray is

$$\frac{10 \pi}{16 \pi }$ \times 100 % = $\frac{5}{8}$ \times 100 % =62 $\frac{1}{2}$ %$ .

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Question

In the above figure, .

Give the ratio of the area of the outer ring to that of the inner circle.

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Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ .

The areas of the largest circle and the second-largest circle are, respectively,

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

The difference of their areas, which is the area of the outer ring, is

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$ .

The inner circle has area

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$ .

The ratio of these areas is therefore

$\frac{7 \pi}{\pi}$ = $\frac{7}{1}$ , or 7 to 1.

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