chapter 10.6. circle a set of all points equidistant from the center
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CIRCLES AND ARCSChapter 10.6
Circle
A set of all points equidistant from the center
Center
Circle
A circle is named by the center
Circle P (P)
P
Diameter
A segment that contains the center of a circle and has both endpoints on the circle.
Diameter
Radius
A segment that has one endpoint at the center of the circle and the other on the circle.
Radius
Congruent Circles
Congruent circles have the congruent radii
P Q
Central Angle
An angle whose vertex is the center of the circle.
Central Angle
Arc
Part of a circle. From point to point on the outside of the circle.
Arc
Semicircle
An arc that’s half of the circle.
SemicircleHas a measure of 1800
1800
Minor Arc
A minor arc is smaller than half the circle.
Minor Arc
400
Same measure as the corresponding interior angle
Major Arc
A major arc is larger than half the circle.
Major Arc360 minus the minor arc
400
3200
Practice 1
Name 3 of the following in A.1. the minor arcs
2. the major arcs
3. the semicircles
Adjacent Arcs Adjacent arcs are arcs of the same circle
that have exactly one point in common.
Arc Addition Postulate The measure of the arc formed by two adjacent
arcs is the sum of the measure of the two arcs.
400 7001100
Practice 2
Find the measure of each arc in R.
1. UT
2. UV
3. VUT
4. ST
5. VS
Practice 3
Find each indicated measure for D.
1. mEDI
2.
3. mIDH
4.
Circumference
The distance around the circle A measure of length
Circumference The circumference of a circle is π times the
diameter (a = πd) or 2 times π and the radius (a = 2πr).
Diameter
Circumference
Example:
D = 4
C = d= 4
or = 12.52
Circumference
Example:
C = 2r= 2(5)
or = 31.4r = 5= 10
Practice 4
Find the circumference of each circle. Leave your answer in terms of .
1. 2.
Arc Length
The length of an arc is calculated using the equation:
600
measure of the arc________________360 * circumference
Arc Length
The length of an arc is calculated using the equation:
600
measure of the arc________________360 * d
Arc Length
The length of an arc is calculated using the equation:
600
measure of the arc________________360 * 2r
Arc Length
________________measure of the arc360 * d
600
7
Arc Length
________________ 60360 * 7
600
7
Arc Length
________________ 1 6 * 22
600
7
= 3.67
Practice 5
Find the length of each darkened arc. Leave your answer in terms of .
1. 2.
Area of a Circle
The product of π and the square of the radius.
A = r2
Radius
Area of a Circle
Example:
A = r2
= 52
or = 78.54r = 5= 25
Practice 6
Find the area of a circle:
1. 6 in. radius
2. 10 cm. radius
3. 12 ft. diameter
Sector of a Circle
A sector of a circle is a region bounded by an arc of the circle and the two radii to the arc’s endpoints.
You name a sector using the two endpoints with the center of the circle in the middle.
Sector of a Circle
Sector is the area of part of the circle
Area of blue section
Area of Sector of a Circle
The area of a sector is:
measure of the arc________________360 * r2
Sector of a Circle
Find the area of the sector
600
12
Arc Length
________________measure of the arc360 * r2
600
12
Arc Length
________________ 60360 * 122
600
12
Arc Length
________________ 1 6 * 144
600
12
= 24
Segment of a Circle
Part of a circle bounded by an arc and the segment joining its endpoints
Area of a Segment of a Circle Equal to the area of the sector minus the
area of a triangle who both use the center and the two endpoints of the segment.
Sector – Triangle = Segment
Area of a Segment of a Circle
- =
Area of a Segment of a Circle Find the area of the segment.
600
12
Area of a Segment of a Circle Separate the triangle and the sector
600
12600
12
Area of a Segment of a Circle Find the area of both figures
600
12600
12
Area of Sector
600
12
________________ 60360 * 122
= 24
Area of Triangle
600
6ð3
Find the altitude 12
or 10.4Find the base
6
Area of Triangle
600
12
10.4
6
a = ½bh
= ½(12)(10.4)
= 62.4
Area of a Segment of a Circle Subtract the triangle from the Sector
24 62.4
-24 62.4 = 13