Transcript Area of a Circle - My Teacher Site

Circumference and Arc Length
Areas of Circles and Sectors
Objectives:
1. To find circumference and arc length
2. To find the area of circles and sectors
Let’s Have Some π
The distance around a circle is its
circumference.
Diameter
Circumference
Circumference of a Circle
The circumference C of
a circle is C = πd or
C = 2πr, where d is
the diameter and r is
the radius of the
circle.
Example 1
A delicious cherry pie
has a radius of 6
inches. How much
crust does that
delightful dessert
have around its
circumference?
Example 2
If a circle has a
circumference of
10π meters, what is
the radius?
?m
Example 3
1. Circumference =
2. Circumference =
3. Radius =
Example 4
A car tire has a diameter of 28 inches. How
many revolutions does the tire make while
traveling 500 feet?
Example 5
Assume the delicious
cherry pie with 6
inches is sliced into
6 equal pieces.
What is the length
of the crust on each
of those pieces?
Arc Measure and Arc Length
The measure of an
arc is the measure
of the central angle
it intercepts. It is
measured in
degrees.
Arc Measure and Arc Length
An arc length is a
portion of the
circumference of a
circle. It is measured
in linear units and
can be found using
the measure of the
arc.
Example 6
Assume the radius of each circle below is 24
units. Find the length of arc AB.
B
B
B
90
60
A
45
A
A
Arc Length Corollary
If the radius of a circle
is r and two radii
form a central angle
of a°, then the length
of the arc formed by
those radii is given
by the formula:
B
r
a
A
Arc Length Corollary
If the radius of a circle
is r and two radii
form a central angle
of a°, then the length
of the arc formed by
those radii is given
by the formula:
B
r
a
A
Example 7
Find the length of arc AB.
1.
2.
3.
Example 8
Find the indicated measure.
1. Circumference =
2. Radius =
Example 9
Find the perimeter of the region.
1.
2.
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Recall that when investigating the area of certain
polygons, we based our new area formulas on
shapes whose area we already knew. We’ll do
the same with the area of a circle. (It’s probably
the coolest thing you’ve ever seen.)
Area of a Circle
Area of a Circle Theorem
The area of a circle is π times the square of
the radius.
Example 10
A cherry pie has a
diameter of 12
inches. What is the
area accurate to the
nearest tenth of a
square inch?
Example 11
If the irrigated circular
field shown has an
area of 804.25 m2,
what is the length
of the rotating
sprinkler (the radius
of the circle)?
Example 12
Find the exact area. Then find the area to
the nearest hundredth of a square unit.
1.
2.
3.
Example 13
1. The area of a circle
is 58 square inches.
Find the radius.
2. The area of a circle
is 106 square
centimeters. Find
the diameter.
Sector: An Actual Slice of Pie
A sector of a circle is
the region between
two radii of a circle
and the included
arc.
Example 14
Remember that delicious
cherry pie from Example
5? Well, it’s mostly a
memory now, as there is
only one slice left! What
is the area of that piece
of pie if the radius is 6
inches and the angle
formed by the sides of
the pie slice is 60°?
Got any vanilla ice
cream for that?
Area of a Sector Conjecture
If the radius of a circle
is r and two radii
form a central angle
of a°, then the area
of the sector formed
by those radii is
given by the formula
a
 r2
360 
Area of a Sector Conjecture
If the radius of a circle
is r and two radii
form a central angle
of a°, then the area
of the sector formed
by those radii is
given by the formula
Example 15
The radius of a circle
is 18 cm. A sector
is formed by a
central angle
measuring 40°.
What is the exact
area of the sector?
Example 16
Find the area of each sector.
1.
2.
Example 17
1. Find the area of
circle S.
2. Find the radius of
circle S.
Example 18
What is the area of
the shaded region if
the radius of each
circle is 6 cm.