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ISEE Upper Level Math : Radius

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Example Question #1 : How To Find The Length Of A Radius

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

Possible Answers:

r=9

r=6

r=18

r=3

Correct answer:

r=18

Explanation:

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

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

$36\pi=2r\pi$

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

36=2r

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

18=r

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Example Question #1 : How To Find The Length Of A Radius

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

Possible Answers:

$3\sqrt{2}$

$2\sqrt{3}$

Correct answer:

$3\sqrt{2}$

Explanation:

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:

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

$r^{2}=18$

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

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Example Question #3 : How To Find The Length Of A Radius

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

Possible Answers:

r=10

r=8

r=12

r=24

r=9

Correct answer:

r=12

Explanation:

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 .

$24\pi=2r\pi$

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

24=2r

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

r=12

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Example Question #4 : How To Find The Length Of A Radius

Inscribed angle

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

Possible Answers:

100

Correct answer:

Explanation:

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

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

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

r = 50

The radius of the circle is 50.

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Example Question #5 : How To Find The Length Of A Radius

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

Possible Answers:

Not enough information to determine.

125 m

384 m^2

384 m

Correct answer:

384 m

Explanation:

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:

384m=r

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Example Question #6 : How To Find The Length Of A Radius

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?

Possible Answers:

26m

13m

Cannot be determined from the information provided

15.5m

Correct answer:

13m

Explanation:

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

169m^2=r^2

Then, square root both sides.

$r=\sqrt{169m^2}=13m$

So our answer is 13m.

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Example Question #1 : Area Of A Circle

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

Possible Answers:

960

153.938

940

706.8583

960

176.7146

940

Correct answer:

176.7146

Explanation:

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 7.5 for in our equation.

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

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Example Question #2 : Area Of A Circle

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

Possible Answers:

$3\pi$

$9\pi$

$18\pi$

$36\pi$

Correct answer:

$9\pi$

Explanation:

First, solve for radius:

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

Then, solve for area:

$A=r^2\pi=3^2\pi=9\pi$

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Example Question #1 : How To Find The Area Of A Circle

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

Possible Answers:

$12.56\ cm^2$

$11.56\ cm^2$

$12 \ cm^2$

$13\ cm^2$

$13.56\ cm^2$

Correct answer:

$12.56\ cm^2$

Explanation:

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 3.14 .

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

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Example Question #4 : Area Of A Circle

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

Possible Answers:

12 t^2

12.56 t^2

11.56 t

11.56 t^2

12.56 t

Correct answer:

12.56 t^2

Explanation:

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 3.14 .

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

$\Rightarrow Area\approx 12.56t^2$

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ISEE Upper Level Math : Radius

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Length Of A Radius

Example Question #1 : How To Find The Length Of A Radius

Example Question #3 : How To Find The Length Of A Radius

Example Question #4 : How To Find The Length Of A Radius

Example Question #5 : How To Find The Length Of A Radius

Example Question #6 : How To Find The Length Of A Radius

Example Question #1 : Area Of A Circle

Example Question #2 : Area Of A Circle

Example Question #1 : How To Find The Area Of A Circle

Example Question #4 : Area Of A Circle

All ISEE Upper Level Math Resources

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