Area of Circle, Sector and Segment

SECONDARY LEVEL

Session #2 Presented by: Dr. Del Ferster

Immaculata Week 2013July 29—August 2, 2013

Today, we’ll look at finding area of a circle, and extending that idea to:◦ Find the area of a sector of a circle.◦ Find the area of a segment of a circle.

We’ll take a look at using trigonometry to help to find the area of a triangle.

Of course, I’ll have some problems for us to solve.

And again, as usual, I’ve collected some handouts that you’re welcome to have.

What’s in store for today?

Let’s see what you remember about area and

circles.

The area of a circle is times the square of the radius or A = r2

Of course, some problems might try to be tricky and give you the diameter, or maybe even the circumference of the circle.

Area of a Circle

2CircleA r

A SECTOR of a circle is a part of a circle comprised by two radii and the arc that those radii intercept.

We can name a sector by using the endpoint of one radius, the center of the circle, and the endpoint of the other radius.

For those of you who remember trivial pursuit, it’s the shape of the pieces you tried to earn by answering questions.

What is a SECTOR of a circle?

arc

radiu

s

radius BO

Sector AOB of Circle O

A

The area of a sector is a fractional part of the area of a circle.

The ratio of a sector’s area to a circle’s area is:

More on Sectors

Measure of the central angle of the sector360

Area of a Sector Formula

m˚360 πr2Area of a sector =

measure of the central angle or arc

The fraction of the circle

The area of the entire circle

.

Let’s look at some problems

We’ll focus on circles and sectors for now.

Area of a circle

Area of the sector, with a central angle of m degrees. (think of this as a fractional portion of the entire circle)

Recall the following facts

2A r

2

360mA r

The radius of the circle is 10 cm.

Angle AOB is 450

What is the area of the sector AOB? Leave your answer in terms of

Example 1

252

Use the diagram shown to find the following. Leave your answers in terms of

1. The area of the circle.

2. The area of the red sector.

3. The area of the blue sector

Example 2

196196

3

3923

Bob is hungry for pizza, however, due to the sequester, the radius of the pizzas have been reduced to 6 inches. Darn sequester!!

Bob’s slice of the pizza has an area of

HOW MANY SLICES WAS THE PIZZA CUT INTO?

What was the size of the central angle for Bob’s slice of pizza?

Example 3

3

The area of sector AOB is

The measure of

Find the length of the radius of circle O.

Example 4

A O

B

94

90oAOB

The area of the BIG SECTOR (the yellow one) is

The measure ofthe central angle for the purple sector is

Find the length of the radius of circle O.

Example 5

A O

B

48

AOB

90o

Let’s take a look at SEGMENTS

We’ll start with a definition and look at some problems.

A segment of a circle is a region bounded by an arc and its chord

A SEGMENT of a Circle

For those of you who enjoy pie, an alternate look at Area of a

SegmentA segment is the portion of the circle

with the radii connected at their endpoints and the central arc.

(The crust of the pie) slice.)

radius

radius

The red section is

the segment.

To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

Finding the area of a segment

Unfortunately, finding the area of the triangle might not be as easy as you might think.

Remember that when we use the formula to find the area of a triangle.

Base and height MUST be perpendicular to each other!!

Doesn’t sound too bad, does it?

12triangleA bh

Example 1: An easy one

Find the area of a segment of a circle whose radius is 10 cm, and central angle is 90°.

Step 1: Find the area of the sector.

Step 2: Find the area of the triangle.Area = ½ bh

Step 3: Find the area of the segment.

radius

radius

2

360mArea r

Yep, the measure of the central angle was 90 degrees, that made the length of the radius applicable for both base and height for our triangle.

Sadly, when the measure of the central angle isn’t 90 degrees, it’s more of a pain to calculate the area of the triangle.

Why was that one easy?

Find the area of segment LNM to the nearest hundredth.

Step 1 Find the area of sector LNM.

Example #2: A bit tougher

12

2

360mArea r

6 cm

Let’s take a closer look at that triangle. Warning, we’re about to use 30-60-90 triangle ratios.Step 2: Find the area of ∆LNM. Draw altitude--h

Example #2: Continued

6 612

A bh

9 3

Step 3: area of segment =

area of sector LNM – area of ∆LNM

Example #2: Concluded

12 9 3

6 cm

cm

That was tough!Is there another way to find the area of the triangle?

YIKES…that was about as much fun as a root canal.

We can make use of trigonometry to find the area of the triangle formed by 2 radii with a given central angle measure, let’s call it

Finding the area of a triangle knowing 2 sides (both of length r) and the included angle,

Here’s a link, if you’re so inclined, that explains the derivation of the formula

Yes, but it might not be as pretty as you wished.

21 sin2triangleA r

SHOW ME THAT WORK

Find the area of a segment of a circle whose diameter is 24 inches and whose central angle is 20 degrees.

Grab your pencil, we’ll do this one on the chalkboard.

Let’s do one that way. Shall we?

21 sin2triangleA r 2

360mArea r

PROBLEMSGrab your pens and pencils, and let’s do some math!Feel free to work by yourself or with a partner.

Find the area of the shaded region. Leave your answer in terms of

Problem #13. Find the area of the shaded region.

The radius of the circle is 12 in.

What is the area of sector ACB. Leave your answer in terms of

Problem #2

120oACB

Find the area of the indicated segment. Leave your answer in terms of

Problem #3

Find the area of the red portion of the sign. Leave your answer in terms of

Problem #4

The circle graph shows the results of a survey about the favorite types of muffins. Find the central angle measure of the sector for each type of muffin.

Explain your solution.

Problem #5

Find the area of the record. THAT’s the black part, for you ITUNES people.

Problem #6

1cm3cm

5cm

Find the shaded area if r=10. (Note: r is the length of the radius of the small circles.) Leave your answer in terms of

Problem #7

Find the area of sector ACB. Leave your answer in terms of

Problem #8

Find the area of the shaded segment. Round your final answer to the nearest hundredth.

Problem #9

Find the area of the shaded region. Point O is the center of the circle, and the radius is 4 units.

Problem #10

430

O

The rectangular wall shown above has an entrance cut into it. You want to paint the wall. To the nearest square foot, what is the area of the region that you need to paint? If one gallon of paint will cover 200 square feet of surface area, how many gallons of paint will you need to do the project?

Problem #11

The diagram shown above, models a rectangular rear window of a car. This fancy car has a rear window wiper that can rotate through 160 degrees. What percentage of the rear window is cleaned by the wiper.

Problem #12

160º

20 cm

40 cm

Area c leaned by the w ipe r

132 cm

72 cm

ANSWERS TO PROBLEMS Problem Answer

12345

BlueberryBanana Nut

BranOther

108 2748

4 8 21

144o

126o

54o36o

ANSWERS TO PROBLEMS (Continued)

Problem Answer67891011

12

6520010

353.778 8 3

2579.469 ft3 gallons47.012%

Questions or comments

Remember, the process is the most important thing. Don’t memorize, UNDERSTAND!

Challenge your students, support them, and share their successes!!

Wrapping it up for today?

Thanks for your attention, participation, and energy. (I know it’s a long time to sit! )

Grab your shades and your beach gear and head out!

If I can be of help during the school year, please don’t hesitate to let me know

EMAIL:◦ Here at Immaculata:◦ dferster@immaculata.edu◦ My home email:◦ delferst@gmail.com

PHONE: (610) 369-7344 (HOME) (610) 698-7615 (CELL)

The last slide!!!