Call Now to Set Up Tutoring:

(888) 888-0446

High School Math : Sectors

Study concepts, example questions & explanations for High School Math

Create An Account

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : How To Find The Area Of A Sector

Find the area of the shaded region:

Screen_shot_2014-02-27_at_6.35.30_pm

Possible Answers:

$108\pi-108\sqrt{3}m^2$

$108\pi-81\sqrt{3}m^2$

$81\pi-81\sqrt{3}m^2$

$108\pi-81\sqrt{2}m^2$

$81\pi-108\sqrt{3}m^2$

Correct answer:

$108\pi-81\sqrt{3}m^2$

Explanation:

To find the area of the shaded region, you must subtract the area of the triangle from the area of the sector.

The formula for the shaded area is:

$A=A_{sector}-A_{triangle}$

$A = \pi(r^2)(part)-\frac{1}{2}(b)(h)$ ,

where is the radius of the circle, part is the fraction of the sector, is the base of the triangle, and is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a 30-60-90 triangle:

$a-a\sqrt{3}-2a$ , where is the side opposite the $\measuredangle30$ .

Plugging in our final values, we get:

$A = \pi(18m^2)(\frac{120}{360})-\frac{1}{2}(18\sqrt{3}m)(9m)$

$A = 108\pi-81\sqrt{3}m^2$

Report an Error

Example Question #2 : How To Find The Area Of A Sector

Find the area of the following sector:

Possible Answers:

$3 \pi\ m^2$

$6 \pi\ m^2$

$12 \pi\ m^2$

$18 \pi\ m^2$

$9 \pi\ m^2$

Correct answer:

$9 \pi\ m^2$

Explanation:

The formula for the area of a sector is

$A = \pi r^2 (part)$ ,

where is the radius of the circle and part is the fraction of the sector.

Plugging in our values, we get:

$A = \pi (6\ m)^2 (\frac{90}{360})$

$A=9 \pi\ m^2$

Report an Error

Example Question #1 : How To Find The Area Of A Sector

Circle

The radius of the circle above is and $\angle A=45^{\circ}$ . What is the area of the shaded section of the circle?

Possible Answers:

$4\pi$

$\pi$

$2\pi$

$8\pi$

$16\pi$

Correct answer:

$2\pi$

Explanation:

Area of Circle = πr² = π4²= 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

Report an Error

Example Question #11 : Circles

Find the area of the shaded segment of the circle. The right angle rests at the center of the circle.

Possible Answers:

$9\pi+36$

$9\pi$

$9\pi-36$

$36\pi-18$

$9\pi-18$

Correct answer:

$9\pi-18$

Explanation:

We know that the right angle rests at the center of the circle; thus, the sides of the triangle represent the radius of the circle.

r=6

Because the sector of the circle is defined by a right triangle, the region corresponds to one-fourth of the circle.

$\frac{degrees\ in\ angle}{degrees\ in\ circle}=\frac{90^o}{360^o}=\frac{1}{4}$

First, find the total area of the circle and divide it by four to find the area of the depicted sector.

$\small A(circle)=\pi(r)^2$

$\small A(sector)=\frac{\pi(r)^2}{4}$

$\small A(sector)=\frac{\pi(6)^2}{4}=\frac{\pi(36)}{4}=9\pi$

Next, calculate the area of the triangle.

$\small A(tri)=\frac{1}{2}bh=\frac{1}{2}(6)(6)=18$

Finally, subtract the area of the triangle from the area of the sector.

$\small A(shaded)=9\pi-18$

Report an Error

Example Question #1 : How To Find The Area Of A Sector

Square-missing

SRAD is a square.

$\overline{AB}=\overline{BC}=\overline{CD}$

The arc from to is a semicircle with a center at the midpoint of $\overline{BC}$ .

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot.

In dollars, what is the total upkeep cost for the summer?

Possible Answers:

$\$(2250-31.25\pi )$

$\$900$

$\$(2250-250\pi )$

$\$(900-12.5\pi )$

$\$(900-25\pi )$

Correct answer:

$\$(2250-31.25\pi )$

Explanation:

To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle.

The area of the square is straightforward:

30 * 30 = 900 square feet

Because each side is 30 feet long, AB + BC + CD = 30.

We can substitute BC for AB and CD since all three lengths are the same:

BC + BC + BC = 30

3BC = 30

BC = 10

Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet.

The area of the semi-circle is half the area of a circle with radius 5. The area of the full circle is 5²π = 25π, so the area of the semi-circle is half of that, or 12.5π.

The total area of the plot is the square less the semicircle: 900 - 12.5π square feet

The cost of upkeep is therefore 2.5 * (900 – 12.5π) = $(2250 – 31.25π).

Report an Error

Example Question #2 : How To Find The Area Of A Sector

In the figure, PQ is the arc of a circle with center O. If the area of the sector is $3\pi$ what is the perimeter of sector?

Possible Answers:

$3 + 2\pi$

$12 + \pi$

$6 + \pi$

$12 + 2\pi$

$1 + \pi$

Correct answer:

$12 + \pi$

Explanation:

First, we figure out what fraction of the circle is contained in sector OPQ: $\frac{30^{\circ}}{360^{\circ}}= \frac{1}{12}$ , so the total area of the circle is $\dpi{100} \small 12\times 3\pi=36$ .

Using the formula for the area of a circle, ${\pi}r^{2}$ , we can see that $\dpi{100} \small r=6$ .

We can use this to solve for the circumference of the circle, $2{\pi}r$ , or $12{\pi}$ .

Now, OP and OQ are both equal to r, and PQ is equal to $\dpi{100} \small \frac{1}{12}$ of the circumference of the circle, or ${\pi}$ .

To get the perimeter, we add OP + OQ + PQ, which give us $12+{\pi}$ .

Report an Error

Example Question #1 : How To Find The Length Of An Arc

If a quarter of the area of a circle is $9\pi$ , then what is a quarter of the circumference of the circle?

Possible Answers:

$3\pi$

$9\pi$

$12\pi$

$2.25\pi$

Correct answer:

$3\pi$

Explanation:

If a quarter of the area of a circle is $9\pi$ , then the area of the whole circle is $36\pi$ . This means that the radius of the circle is 6. The diameter is 12. Thus, the circumference of the circle is $12\pi$ . One fourth of the circumference is $3\pi$ .

Report an Error

Example Question #1 : How To Find The Length Of An Arc

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Possible Answers:

36 + 10π

18 + 36π

36 + 20π

36 + 36π

18 + 10π

Correct answer:

36 + 10π

Explanation:

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

Report an Error

Example Question #2 : How To Find The Length Of An Arc

Find the arc length of a sector that has an angle of 120 degrees and radius of 3.

Possible Answers:

$36\Pi$

$2\Pi$

$4\Pi$

$16\Pi$

$6\Pi$

Correct answer:

$2\Pi$

Explanation:

The equation for the arc length of a sector is $(\frac{x}{360})2\Pi r$ .

Substitute the given radius for and the given angle for to get the following equation:

$(\frac{120}{360})2\Pi(3)$

Simplify:

$=(\frac{1}{3})2\Pi (3)$

$=2\Pi$

Report an Error

Example Question #2 : How To Find The Length Of An Arc

$\textup{What is the length, in inches, of an arc with a central angle of }$

$72\textup{ degrees on a circle with a radius of 20 inches?}$

Possible Answers:

$10\pi$

$20\pi$

$4\pi$

$8\pi$

$12\pi$

Correct answer:

$8\pi$

Explanation:

$\frac{\textup{length of arc}}{\textup{circumference}}=\frac{\textup{central angle}}{360}$

$\textup{Find circumference: }C=2\pi r = 2\pi 20 = 40\pi$

$\frac{x}{40\pi }= \frac{72}{360}\:\:\textup{reduce: }\frac{x}{40\pi }=\frac{1}{5}\:\textup{ cross-multiply: }\:5x=40\pi$

$x=8\pi$

Report an Error

View Tutors

Deborah
Certified Tutor

Saint Mary's College, Bachelor of Science, Organizational Leadership. Solano Community College, Associate in Science, Busines...

View Tutors

Taye
Certified Tutor

Vanderbilt University, Bachelor of Science, Nursing (RN). Oklahoma City University, Juris Doctor, Legal Studies.

View Tutors

Mudassara
Certified Tutor

CUNY College of Staten Island, Master of Arts, Business Administration and Management.

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GMAT Tutors in Atlanta, SAT Tutors in Miami, ACT Tutors in Chicago, Calculus Tutors in Seattle, ACT Tutors in Houston, LSAT Tutors in San Francisco-Bay Area, GMAT Tutors in Washington DC, Calculus Tutors in San Diego, MCAT Tutors in Los Angeles, Calculus Tutors in Chicago

Popular Courses & Classes

GRE Courses & Classes in Los Angeles, Spanish Courses & Classes in Denver, LSAT Courses & Classes in Houston, ISEE Courses & Classes in New York City, GMAT Courses & Classes in Los Angeles, SAT Courses & Classes in Denver, ISEE Courses & Classes in Seattle, LSAT Courses & Classes in Los Angeles, GRE Courses & Classes in Chicago, ISEE Courses & Classes in Chicago

Popular Test Prep

MCAT Test Prep in Dallas Fort Worth, MCAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Chicago, SAT Test Prep in Philadelphia, LSAT Test Prep in Houston, GRE Test Prep in Houston, SSAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Boston, ACT Test Prep in Philadelphia, GMAT Test Prep in Los Angeles

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

High School Math : Sectors

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2 : How To Find The Area Of A Sector

Example Question #2 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Area Of A Sector

Example Question #11 : Circles

Example Question #1 : How To Find The Area Of A Sector

Example Question #2 : How To Find The Area Of A Sector

Example Question #1 : How To Find The Length Of An Arc

Example Question #1 : How To Find The Length Of An Arc

Example Question #2 : How To Find The Length Of An Arc

Example Question #2 : How To Find The Length Of An Arc

All High School Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: