Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Area of a Sector

Study concepts, example questions & explanations for Trigonometry

Create An Account

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Area Of A Sector

Which of the following is the definition for a sector of a circle?

Possible Answers:

A section of a circle which is enclosed by a chord and the chord’s arc

Any section of a circle

A section of a circle which is enclosed by two radii joined at the center of the circle and the arc between the two radii

Any section of a circle that is triangular in shape

Correct answer:

A section of a circle which is enclosed by two radii joined at the center of the circle and the arc between the two radii

Explanation:

Below is an illustration of a sector of a circle. A sector is the area of a circle which has been enclosed by two radii and the arc between them. A sector is not to be confused with a segment of a circle. A segment is when the area enclosed by the chord of a circle and the arc of the chord.

Report an Error

Example Question #2 : Area Of A Sector

Which of the following is the formula for the area of a sector?

Possible Answers:

$A = \theta \pi r^2$

$A = \frac{\theta\pi}{2} r^2$

$A = \frac{\theta}{2} r^2$

$A = 2\theta r^2$

Correct answer:

$A = \frac{\theta}{2} r^2$

Explanation:

When thinking about how to derive the formula for a sector, we must consider the angle of an entire circle. The angle of an entire circle, 360 degrees, is $2\pi$ and we know the area of a circle is $\pi r^2$ .

When considering a sector, this is only a portion of the entire circle, so it is a particular $\theta$ out of the entire $2\pi$ . We can plug this into our area for a circle and it will simplify to the area of a sector.

$\frac{\theta}{2\pi}\pi r^2$

$\frac{\theta}{2} r^2$

Report an Error

Example Question #1 : Area Of A Sector

If a circle has a sector with an angle of $\theta = 60$ and diameter of 4, what is the area of the sector?

Possible Answers:

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$\frac{2\pi}{3}$

$2\pi$

Correct answer:

$\frac{2\pi}{3}$

Explanation:

It is always best to draw a picture in order to visualize the problem you are trying to solve. The figure below shows the sector we are trying to find the area of.

We know that the formula to find the area of a sector is $A = \frac{\theta}{2} r^2$ . From the information given above we know that the diameter is 4. Since we only need the radius for our formula we divide the diameter by 2 to get the radius length. The radius has a length of 2. We also know that we have our angle measure in degrees and must convert it to radians. We use the conversion formula $degrees \frac{\pi}{180} = radians$ .

$60 \frac{\pi}{180} = \frac{\pi}{3}$

Now we can plug everything into our formula and solve.

$A = \frac{\theta}{2}r^2$

$A = \frac{\pi}{3}\frac{1}{2} 2^2$

$A = \frac{\pi}{6}{4}$

$A = \frac{2\pi}{3}$

Report an Error

Example Question #4 : Area Of A Sector

True or False: The formula to find the area of a sector only works for acute angles.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is not true. Even obtuse angles are less than $2\pi$ so this formula will still work. We can demonstrate this using the sector below. The radius of the circle is 6 and the obtuse angle is 330 degrees.

Converting 330 degrees to radians:

$330 \frac{\pi}{180} = \frac{11\pi}{6}$

We can now plug this into our formula

$A = \frac{\theta}{2} r^2$

$A = \frac{11\pi}{6}\frac{1}{2} 6^2$

$A = \frac{11\pi}{12} 36$

$A = \frac{396\pi}{12}$

$A = 33\pi$

Now we can confirm this to be true by computing the area of the sector formed by the

area leftover formed by the acute angle, 30 degrees. To do this we will first find the total

area of the circle and then subtract the area of the sector formed by the acute angle. This should be equal to the area of the larger vector if our formula works for all angles because the sum of both sectors should be the total area of the circle.

To find the area of the circle:

$\pi r^2$

$36\pi$

To find the area of the smaller sector (note, 30 degrees in radians is $\frac{\pi}{6}$ :

$A = \frac{\theta}{2} r^2$

$A = \frac{\pi}{6}\frac{1}{2} 6^2$

$A = \frac{\pi}{12} 36$

$A = 3\pi$

Clearly, the total area of the circle minus the area of the small sector is equal to the area

of the larger circle, therefore this formula works for all angles less than $2\pi$

$36\pi - 3\pi = 33\pi$

Report an Error

Example Question #5 : Area Of A Sector

Knowing that the arc length of a sector $3\pi$ and the angle measure is $\theta = \frac{4\pi}{3}$ , what is the area of the sector?

Possible Answers:

$A = \frac{\pi}{3}$

$A = \frac{81\pi}{127}$

$A = \frac{\pi}{4}$

$A = \frac{27\pi}{8}$

Correct answer:

$A = \frac{27\pi}{8}$

Explanation:

To solve this problem we must know the formula for finding the arc length of a sector. This formula is $l_A = \theta r$ . With the given information we are able to solve for the radius which we can then use to solve for the area of the sector itself.

$l_A = \theta r$

$3\pi = \frac{4\pi}{3} r$

$\frac{9}{4} = r$

Now we can plug this radius into the formula to solve for the area of a sector.

$A = \frac{\theta}{2} r^2$

$A = (\frac{4\pi}{3})(\frac{1}{2})( \frac{9}{4})^2$

$A = (\frac{2\pi}{3})(\frac{81}{16})$

$A = \frac{162\pi}{48}$

$A = \frac{27\pi}{8}$

Report an Error

Example Question #1 : Area Of A Sector

True or False: You have a sector of a circle and are given the radius of the circle and the total area of the circle. You are able to find the area of the sector.

Possible Answers:

True

False

Correct answer:

False

Explanation:

The formula for finding the area of a circle is $A = \pi r^2$ . If we know the radius and the area of the entire circle, we still do not know the angle that forms the sector or the arc length. Therefore we do not have enough information to solve for the sector area. In order to find the area of the sector we need to either know the angle of the sector and the radius, or have some means to solve for this information.

Report an Error

Example Question #7 : Area Of A Sector

A circle has a sector formed by radii of length 3 and an angle of $\frac{5\pi}{4}$ . What is the area of the sector?

Possible Answers:

$A = \frac{9\pi}{8}$

$A = \frac{9\pi}{4}$

$A = \frac{45\pi}{8}$

$A = \frac{8\pi}{5}$

Correct answer:

$A = \frac{45\pi}{8}$

Explanation:

The formula to solve for the area of a sector is $A = \frac{\theta}{2} r^2$ . We have all the information needed to plug the values right into the formula and solve for the area.

$A = \frac{\theta}{2} r^2$

$A = (\frac{5\pi}{4})(\frac{1}{2})(3^2)$

$A = (\frac{5\pi}{8})(9)$

$A = \frac{45\pi}{8}$

Report an Error

Example Question #8 : Area Of A Sector

You are given a circle with a sector that is formed by an arc length of $9\pi$ . The circle has a diameter of 20. What is the area of the sector?

Possible Answers:

$A=\frac{9\pi}{20}$

$A=\frac{9\pi}{4}$

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

$A=9\pi$

Correct answer:

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

Explanation:

First, if we have a diameter of 20, we know that the radius must be half of the diameter. We have a radius length of 10. Now, we know the arc length so we must use the arc length formula to solve for the angle measure. To find the length of an arc of a sector we use the formula $l_A = \theta r$ .

$l_A = \theta r$

$9\pi = \theta 10$

$\frac{9\pi}{10} = \theta$

By solving for the angle, we have enough information to solve for the sector area.

$A = \frac{\theta}{2} r^2$

$A = (\frac{9\pi}{10})(\frac{1}{2})(10^2)$

$A = (\frac{9\pi}{20}(100)$

$A = \frac{900\pi}{200}$

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

Report an Error

View Trigonometry Tutors

Ivory
Certified Tutor

Hopkinsville Community College, Associate, Associates in Science. Western Kentucky University, Bachelor in Arts, Psychology.

View Trigonometry Tutors

Ian
Certified Tutor

Lipscomb University, Bachelor of Science, Biology, General.

View Trigonometry Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

SSAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Miami, SAT Courses & Classes in Denver, LSAT Courses & Classes in Chicago, GRE Courses & Classes in Chicago, SAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Phoenix, ACT Courses & Classes in Miami, MCAT Courses & Classes in Los Angeles, SAT Courses & Classes in Seattle

Popular Test Prep

SSAT Test Prep in Phoenix, GMAT Test Prep in Philadelphia, MCAT Test Prep in Chicago, SSAT Test Prep in Los Angeles, SSAT Test Prep in Houston, SSAT Test Prep in New York City, SSAT Test Prep in Miami, ACT Test Prep in Dallas Fort Worth, SSAT Test Prep in Denver, GRE Test Prep in Phoenix

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Trigonometry : Area of a Sector

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Area Of A Sector

Example Question #2 : Area Of A Sector

Example Question #1 : Area Of A Sector

Example Question #4 : Area Of A Sector

Example Question #5 : Area Of A Sector

Example Question #1 : Area Of A Sector

Example Question #7 : Area Of A Sector

Example Question #8 : Area Of A Sector

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: