Sectors - Intermediate Geometry

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Question

How many degrees are in of a circle?

Answer

There are degrees in a circleso the equation to solve becomes a simple percentage problem:

$x=0.35\cdot 360 = 126$

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Question

A sector contains $\small 12.5%$ of a circle. What is the measure of the central angle of the sector?

Answer

An entire circle is $\small 360^\circ$ . A sector that is $\small 12.5%$ of the circle therefore has a central angle that is $\small 12.5%$ of $\small 360^\circ$ .

$\small 12.5%(360)=45$

Therefore, our central angle is $\small 45^\circ$

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Question

If you have percent of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

Now you need to convert into a decimal.

$66.7% \rightarrow \frac{66.7}{100}=0.667$

If you multiply 360 by 0.667, you get the degree measure that corresponds to the percentage, which is 240.

$360 \cdot 0.667=240$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First convert into a decimal.

$20% \rightarrow \frac{20}{100}=0.2$

If you multiply 360 by 0.20, you get the degree measure that corresponds to the percentage, which is 72.

$360 \cdot 0.2=72$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

In order to start this problem we need to convert the percent into a decimal.

$30% \rightarrow \frac{30}{100}=0.3$

If you multiply 360 by 0.30, you get the degree measure that corresponds to the percentage, which is 108.

$360\cdot 0.3=108$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First convert the percent to decimal.

$35% \rightarrow \frac{35}{100}=0.35$

Now if you multiply 360 by 0.35, you get the degree measure that corresponds to the percentage, which is 126.

$360 \cdot 0.35=126$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First convert the percentage into a decimal.

$90% \rightarrow \frac{90}{100}=0.9$

If you multiply 360 by 0.90, you get the degree measure that corresponds to the percentage, which is 324.

$360 \cdot 0.90=324$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First we need to convert the percentage into a decimal.

$45% \rightarrow \frac{45}{100}=0.45$

If you multiply 360 by 0.45, you get the degree measure that corresponds to the percentage, which is 162.

$360 \cdot 0.45=162$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

In order to solve this problem we first need to convert the percentage into a decimal.

$37.5% \rightarrow \frac{37.5}{100}=0.375$

If you multiply 360 by 0.375, you get the degree measure that corresponds to the percentage, which is 135.

$360 \cdot 0.375=135$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First we need to convert the percentage into a decimal.

$70% \rightarrow \frac{70}{100}=0.7$

If you multiply 360 by 0.70, you get the degree measure that corresponds to the percentage, which is 252.

$360 \cdot 0.7 =252$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First we need to convert the percentage into a decimal.

$80% \rightarrow \frac{80}{100}=0.8$

If you multiply 360 by 0.80, you get the degree measure that corresponds to the percentage, which is 288.

$360 \cdot 0.8= 288$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First convert the percentage into a decimal.

$44% \rightarrow \frac{44}{100}=0.44$

If you multiply 360 by 0.44, you get the degree measure that corresponds to the percentage, which is 158.4.

$360 \cdot 0.44=158.4$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

First convert the percentage into a decimal.

$18% \rightarrow \frac{18}{100}=0.18$

If you multiply 360 by 0.18, you get the degree measure that corresponds to the percentage, which is 64.8.

$360 \cdot 0.18=64.8$

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Question

If you have of a circle, what is the angle, in degrees, that creates that region?

Answer

A full circle has 360 degrees, which means that 100% of the circle is 360 degrees.

We first need to convert the percentage into a decimal.

$56% \rightarrow \frac{56}{100}=0.56$

If you multiply 360 by 0.56, you get the degree measure that corresponds to the percentage, which is 201.6.

$360 \cdot 0.56=201.6$

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Question

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Answer

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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Question

What is the angle of a sector of area on a circle having a radius of ?

Answer

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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Question

What is the angle of a sector that has an arc length of on a circle of diameter ?

Answer

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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Question

What is the sector angle, in degrees, if the area of the sector is $4\pi$ with a given radius of ?

Answer

Write the formula for the area of a circular sector.

$A= \frac{\Theta }{360}\pi r^2$

Substitute the given information and solve for theta:

$4\pi= \frac{\Theta }{360}\pi (4)^2$

$\frac{1}{4}= \frac{\Theta }{360}$

$\Theta= \frac{360}{4} = 90^{\circ}$

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Question

The length of the intercepted arc of a sector of a circle with radius $\small 12$ meters is $\small \small 4\pi$ meters. Find the measure of the central angle of the sector.

Answer

The formula for the length of an arc is

$\small s=\frac{m}{360}2\pi r$

where $\small m$ is measure of the central angle and $\small r$ is the radius. Substituting what we know gives.

$\small \small 4\pi=\frac{m}{360}(2\pi)(12)$

$\small \small 4\pi=\frac{m\pi}{15}$

$\small \small 60\pi=m\pi$

$\small \small m=60$

Therefore, our central angle has a measure of $\small 60^\circ$

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Question

A sector in a circle with a radius of has an area of $12\pi$ . In degrees, what is the measurement of the central angle for this sector?

Answer

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$

The central angle is degrees.

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