Circles - GED Math

Card 0 of 890

Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

The area of the shaded sector is $24 \pi$ . The area of the white sector is $40 \pi$ .

What is the length of arc $\widehat{MN}$ ?

Answer

The area of the circle is the sum of the areas of the sectors, which is

$A = 24 \pi + 40 \pi = 64 \pi$ .

The degree measure of the arc of the shaded sector is

$\frac{24 \pi}{64 \pi} \times 360 ^{\circ } = 135 ^{\circ }$ .

The radius can be found by solving for and substituting $A = 64 \pi$ in the area formula:

$A = \pi r^{2}$

$64 \pi = \pi r^{2}$

$r^{2} = 64$

The length of arc $\widehat{MN}$ is

$\frac{135}{360} \cdot 2 \pi r = \frac{135}{360} \cdot 2 \cdot 8 \pi = 6\pi$ .

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram.

The length of arc $\widehat{MN}$ is $12 \pi$ .

The length of arc $\widehat{MYN}$ is $18 \pi$ .

What is the area of the white sector?

Answer

The circumference of the circle is the sum of the lengths of the arcs, which is

$C = 12 \pi + 18 \pi = 30 \pi$ .

The white sector has degree measure

$\frac{ 18 \pi }{ 30 \pi} \times 360 ^{\circ} = 216 ^{\circ}$ .

The radius of the circle can be found using the circumference formula, setting $C = 30 \pi$ :

$C = 2 \pi r$

$30 \pi = 2 \pi r$

$r = 30 \pi \div 2 \pi$

The area of the white sector is therefore

$A = \frac{216}{360 } \cdot \pi r^{2} = 0.60 \cdot \pi \cdot 15^{2} = 225 \cdot 0.60 \cdot \pi = 135 \pi$ .

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Question

What percent of the above circle has not been shaded in?

Answer

There are a total of 360 degrees to a complete circle. The shaded sector has degree measure $80^{\circ }$ , so the unshaded sector has degree measure

$360 ^{\circ }- 80^{\circ }= 2 80^{\circ }$

Also, a sector of $N^{\circ }$ is $\frac{N}{360} \times 100 %$ of the circle, so, setting , we find that the unshaded sector is

$\frac{280}{360} \times 100 %$

$= \frac{28,000}{360} %$

of the circle. This reduces to

$\frac{28,000\div 40 }{360 \div 40 } %$

$=\frac{700 }{9 } %$

$=77\frac{7}{9 } %$ ,

the correct percentage.

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Question

What percent of the above circle has been shaded?

Answer

There are a total of $360^{\circ }$ in a circle. The unshaded portion of the circle represents a $140^{\circ }$ sector, so the shaded portion represents a sector of measure

$360^{\circ } - 140^{\circ } = 220 ^{\circ }$ .

This sector represents

$\frac{ 220 ^{\circ }}{360^{\circ } } \times 100 %$

$= \frac{ 22000}{360 } %$

$= \frac{ 22000 \div 40 }{360 \div 40 } %$

$= \frac{ 550 }{9 } %$

$= 61\frac{1}{9} %$

of the circle.

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Question

What percentage of a circle is covered by a sector with a central angle of $240^{\circ}$ ?

Answer

What percentage of a circle is covered by a sector with a central angle of $240^{\circ}$ ?

To find the percentage of a circle from the central angle, we need to use the following formula:

$%=\frac{\theta}{360^{\circ}}100$

Where theta is our central angle.

Plug in our given degree measurement and simplify.

$%=\frac{240^{\circ}}{360^{\circ}}100=66.67 %$

So, our answer is 66.67%

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Question

Figure is not drawn to scale.

What percent of the circle has not been shaded?

Answer

The total number of degrees in a circle is $360^{\circ }$ , so the shaded sector represents $\frac{144^{\circ }}{360^{\circ }}$ of the circle. In terms of percent, this is

$\frac{144^{\circ }}{360^{\circ }} \times 100 % = \frac{14,400^{\circ }}{360^{\circ }} %= 40%$ .

The shaded sector is 40% if the circle, so the unshaded sector is of the circle.

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Question

Find the length of the minor arc if the circle has a circumference of $81\pi$ .

Answer

Recall that the length of an arc is a proportion of the circumference, just like how the measure of a central angle is a proportion of the total number of degrees in a circle.

Thus, we can write the following equation to solve for arc length.

$\text{Arc length}=\frac{\text{central angle}}{360}(\text{circumference})$

Plug in the given central angle and circumference to find the length of the minor arc .

$\text{Arc length}=\frac{40}{360}(81\pi)=9\pi$

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Question

A circle has area $81 \pi$ . Give the length of a $90 ^{\circ }$ arc of the circle.

Answer

The radius of a circle, given its area , can be found using the formula

$A = \pi r^{2}$

Set $A= 81 \pi$ :

$\pi r^{2} = 81 \pi$

Find by dividing both sides by $\pi$ and then taking the square root of both sides:

$\pi r^{2} \div \pi = 81 \pi \div \pi$

$r^{2} = 81$

$r = \sqrt{81} = 9$

The circumference of this circle is found using the formula

$C = 2 \pi r$ :

$C = 2 \pi (9)$

$C= 18 \pi$

A $90 ^{\circ }$ arc of the circle is one fourth of the circle, so the length of the arc is

$L = \frac{1}{4}C$

$= \frac{1}{4} \cdot 18 \pi$

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

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Question

If a sector covers $\frac{4}{5}$ of the area of a given circle, what is the measure of that sector's central angle?

Answer

If a sector covers $\frac{4}{5}$ of the area of a given circle, what is the measure of that sector's central angle?

While this problem may seem to not have enough information to solve, we actually have everything we need.

To find the measure of a central angle, we don't need to know the actual area of the sector or the circle. Instead, we just need to know what fraction of the circle we are dealing with. In this case, we are told that the sector represents four fifths of the circle.

All circles have 360 degrees, so if our sector is four fifths of that, we can find the answer via the following.

$\measuredangle =\frac{4}{5}(360^{\circ})=288^{\circ}$

So our answer is:

$288^{\circ}$

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Question

What is the diameter of a circle with a radius of 9?

Answer

The diameter of a circle is twice the radius:

Plug in the radius value:

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Question

What is the radius of a circle given the diameter is 22?

Answer

The radius is half of the diameter, or 11.

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Question

What is the radius, in inches, of a circle with a diameter of 12 inches?

Answer

The radius is half of the diameter:

From here we plug in our diameter measure of 12 and solve for the radius:

$\frac{12}{2}=r$

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Question

What is the diameter of a circle with radius of 10 inches?

Answer

The diameter is twice the radius:

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Question

What is the radius of a circle that has a circumference of $36\pi$ ?

Answer

The equation for circumference of a circle is as follows:

$C=2 \pi r$

Plug in the given values and solve for the radius:

$36\pi=2\pi r$

Now we divide each side by $2\pi$ :

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Question

Find the diameter of a circle that has a circumference of $25\pi$ ?

Answer

We know the equation for the circumference of a circle is $C=2\pi r$ . We also know that the the diameter is the twice the radius, so we can write $C=D\pi$ .

Plug in the given values and solve:

$25\pi=D\pi$

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Question

What is the diameter of a circle with an area of $144\pi$ ?

Answer

In order to solve this question, start by finding the radius. Recall that the area of a circle is computed as:

$A=\pi r^2$

For our data, this is:

$144\pi = \pi r^2$

Solving for , we get:

or

Now, recall that the diameter is double the radius. Thus, it is .

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Question

What is the radius of a circle with a circumference of ?

Answer

Recall that the circumference formula is:

$C=2\pi r$

Now, for your data, this gives you:

$17=\pi r$

You need to be careful, for some students get confused when they have to divide by $\pi$ . However, just treat it like any other number or variable. This means that you can solve for and get:

$r=\frac{17}{\pi}$

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Question

What is the diameter of a circle given the area of 30?

Answer

Write the area formula of a circle.

$A=\pi r^2$

Substitute the area.

$30=\pi r^2$

Divide by pi on both sides.

$\frac{30}{\pi}=\frac{\pi r^2}{\pi}$

$r^2 = \frac{30}{\pi}$

Square root both sides.

$\sqrt{r^2} =\sqrt{ \frac{30}{\pi}}$

$r=\sqrt{ \frac{30}{\pi}}$

The radius is: $\sqrt{\frac{30}{\pi}}$

The diameter is double the radius.

The answer is: $2\sqrt{\frac{30}{\pi}}$

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Question

What is the diameter of a circle if the area is $14\pi$ ?

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the area into the equation.

$14\pi=\pi r^2$

Divide by pi on both sides.

$\frac{14\pi}{\pi}=\frac{\pi r^2}{\pi}$

$r=\sqrt{14}$

The diameter is twice the radius.

The answer is: $2\sqrt{14}$

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Question

If the diameter of a circle is 12 feet, what is the radius in yards?

Answer

There are three feet in one yard.

Using dimensional analysis, we can determine the given diameter in yards first.

$12 \textup{ ft}(\frac{1 \textup{ yd}}{3 \textup{ ft}}) = 4 \textup{ yds}$

The radius is half the diameter.

The answer is: $2 \textup{ yds}$

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