How to find the angle of a sector

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Math › How to find the angle of a sector

Questions 1 - 10

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

$90^{\circ}$

$0^{\circ}$

$45^{\circ}$

$360^{\circ}$

$60^{\circ}$

Explanation

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:

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

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

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

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?

$5.625^\circ$

$2.775^\circ$

$6.225^\circ$

$8.555^\circ$

Explanation

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.

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?

$30^\circ$

$45^\circ$

$60^\circ$

$15^\circ$

Explanation

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.

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?

$270^\circ$

$285^\circ$

$255^\circ$

$240^\circ$

Explanation

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.

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?

$140^\circ$

$150^\circ$

$130^\circ$

$120^\circ$

Explanation

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.

$\bigtriangleup ABC$ is inscribed in a circle. $\overarc {ABC }$ is a semicircle. $m \angle A = 40 ^{\circ }$ .

Which is the greater quantity?

(a) $m \overarc {BAC}$

(b) $m \overarc {BCA }$

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Explanation

The figure referenced is below:

Inscribed angle

$\overarc {ABC }$ is a semicircle, so $\overarc {ADC }$ is one as well; as a semicircle, its measure is $180 ^{\circ }$ . The inscribed angle that intercepts this semicircle, $\angle B$ , is a right angle, of measure $90 ^{\circ }$ . $m \angle A = 40 ^{\circ }$ , and the sum of the measures of the interior angles of a triangle is $180 ^{\circ }$ , so

$m \angle C = 180 ^{\circ } - (m \angle A + m \angle B )$

$= 180 ^{\circ } - (40 ^{\circ } +90 ^{\circ } )$

$= 180 ^{\circ } - 130 ^{\circ }$

$= 50 ^{\circ }$

$\angle C$ has greater measure than $\angle A$ , so the minor arc intercepted by $\angle C$ , which is $\overarc {BA}$ , has greater measure than that intercepted by $\angle A$ , which is $\overarc {BC}$ . It follows that the major arc corresponding to the latter, which is $\overarc {BA C}$ , has greater measure than that corresponding to the former, which is $\overarc {BC A }$ .

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?

$350^\circ$

$340^\circ$

$330^\circ$

$320^\circ$

Explanation

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.

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?

$10^\circ$

$5^\circ$

$15^\circ$

$20^\circ$

Explanation

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.

Circlesectorgeneral9

The arc-length for the shaded sector is . What is the value of , rounded to the nearest hundredth?

Explanation

Remember that the angle for a sector or arc is found as a percentage of the total degrees of the circle. The proportion of to is the same as to the total circumference of the circle.