Areas of Circles and Sectors Objectives To find circumference and arc length To find the area of circles and sectors Lets Have Some π Circumference The distance around a circle is its ID: 737427
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Slide1
Circumference and Arc LengthAreas of Circles and Sectors
Objectives:
To find circumference and arc length
To find the area of circles and sectorsSlide2
Let’s Have Some π
Circumference
The distance around a circle is its
circumference
. Slide3
Circumference of a Circle
The circumference
C
of a circle is
C = π
d or C = 2
π
r
, where d is the diameter and r is the radius of the circle.Slide4
Example 1
A delicious cherry pie has a radius of 6 inches. How much crust does that delightful dessert have around its circumference?Slide5
Example 2If a circle has a circumference of 10
π
meters, what is the radius?Slide6
Example 3
Circumference
=
Circumference
=
Radius
=Slide7
Example 4A car tire has a diameter of 28 inches. How many revolutions does the tire make while traveling 500 feet?Slide8
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?Slide9
Arc Measure and Arc LengthThe
measure of an arc
is the measure of the central angle it intercepts. It is measured in degrees.Slide10
Arc Measure and Arc LengthAn
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.Slide11
Example 6Assume the radius of each circle below is 24 units. Find the length of arc
AB
.Slide12
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:Slide13
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:Slide14
Example 7Find the length of arc AB.
Slide15
Example 8Find the indicated measure.
Circumference
=
Radius =Slide16
Example 9Find the perimeter of the region.
Slide17
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.)Slide18
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.)Slide19
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.)Slide20
Area of a CircleSlide21
Area of a Circle TheoremThe area of a circle is
π
times the square of the radius
.Slide22
Example 10A cherry pie has a diameter of 12 inches. What is the area accurate to the nearest tenth of a square inch?Slide23
Example 11If the irrigated circular field shown has an area of 804.25 m
2
, what is the length of the rotating sprinkler (the radius of the circle)?Slide24
Example 12Find the exact area. Then find the area to the nearest hundredth of a square unit.
Slide25
Example 13The area of a circle is 58 square inches. Find the radius.
The area of a circle is 106 square centimeters. Find the diameter.Slide26
Sector: An Actual Slice of PieA
sector
of a circle is the region between two radii of a circle and the included arc.Slide27
Example 14Remember 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?Slide28
Area of a Sector ConjectureIf 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 formulaSlide29
Area of a Sector ConjectureIf 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 formulaSlide30
Example 15The 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?Slide31
Example 16Find the area of each sector.
Slide32
Example 17Find the area of circle S.
Find the radius of circle S.Slide33
Example 18What is the area of the shaded region if the radius of each circle is 6 cm.