How to find the length of a radius - ISEE Upper Level Quantitative Reasoning

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

Tap to reveal answer

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 $18\pi$ ?

Tap to reveal answer

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 $24\pi$ ?

Tap to reveal answer

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.

Tap to reveal answer

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?

Tap to reveal answer

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?

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 $18\pi$ ?

Tap to reveal answer

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 $24\pi$ ?

Tap to reveal answer

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.

Tap to reveal answer

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?

Tap to reveal answer

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:

← Didn't Know|Knew It →

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?

Tap to reveal answer

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.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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}$

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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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