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:◦ [email protected]◦ My home email:◦ [email protected]
PHONE: (610) 369-7344 (HOME) (610) 698-7615 (CELL)
The last slide!!!
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