How to find the angle of a sector - Intermediate Geometry

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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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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 of the 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.

$\text{Measurement of Central Angle}=\frac{360\times12\pi}{\pi\times 4^2}=\frac{4320\pi}{16\pi}=270$

The central angle is degrees.

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Question

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

$\text{Measurement of Central Angle}=\frac{360\times \pi}{\pi\times 6^2}=\frac{360\pi}{36\pi}=10$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $2\pi$ . In degrees, what is the measurement of the central angle in the 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.

$\text{Measurement of Central Angle}=\frac{360\times 2\pi}{\pi\times 5^2}=\frac{720\pi}{25\pi}=28.8$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $60\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 60\pi}{\pi\times 8^2}=\frac{21600\pi}{64\pi}=337.5$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $3\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 10^2}=\frac{1080\pi}{100\pi}=10.8$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $8\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 8\pi}{\pi\times 16^2}=\frac{2880\pi}{256\pi}=11.25$

The central angle is degrees.

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Question

A sector in a circle with a radius of has a area of $50\pi$ . In degrees, what is the measurement of the central angle in the 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.

$\text{Measurement of Central Angle}=\frac{360\times 50\pi}{\pi\times 20^2}=\frac{18000\pi}{200\pi}=45$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $48\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 48\pi}{\pi\times 12^2}=\frac{17280\pi}{144\pi}=120$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $\frac{1}{2}\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{2}\pi}{\pi\times 3^2}=\frac{180\pi}{9\pi}=20$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $45\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 45\pi}{\pi\times 12^2}=\frac{16200\pi}{144\pi}=112.5$

The central angle is degrees.

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Question

A sector of a circle with a radius of has an area of $3\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 5^2}=\frac{1080\pi}{25\pi}=43.2$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $\frac{1}{4}\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{4}\pi}{\pi\times 4^2}=\frac{90\pi}{16\pi}=5.625$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $14\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 14\pi}{\pi\times 6^2}=\frac{5040\pi}{36\pi}=140$

The central angle is degrees.

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Question

A sector in a circle with a radius of has an area of $35\pi$ . In degrees, what is the measurement of the central angle of the 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.

$\text{Measurement of Central Angle}=\frac{360\times 35\pi}{\pi\times 6^2}=\frac{12600\pi}{36\pi}=350$

The central angle is degrees.

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