Section 7-6 Circles and Arcs
SPI 32B: Identify central angles of circles given a diagram
SPI 33A: Solve problems involving the properties of arcs
Objectives:
• Find measures of central angles and arcs
• Find circumference and arc length
Geometric
Vocabulary
Of a
Circle
Vocabulary of a Circle
Circle:
• set of all points equidistant from the center
• named by its center (⓪P)
Radius:
• segment that has one endpoint at the center and the
other on the circle ( PC )
Diameter:
• segment that contains the center of the circle and
has both endpoints on the circle ( AB is the diameter)
More Vocabulary
Congruent circles:
• circles that have congruent radii
Central Angle:
• angle whose vertex is the center of the circle
• CPA is a central angle
Circles and Arcs
A researcher surveyed 2000 members of a club to find their ages. The
graph shows the survey results. Find the measure of each central
angle in the circle graph.
Because there are 360° in a circle,
multiply each percent by 360 to
find the measure of each central angle.
65+ : 25% of 360 = 0.25 • 360 = 90
45–64: 40% of 260 = 0.4 • 360 = 144
25–44: 27% of 360 = 0.27 • 360 = 97.2
Under 25: 8% of 360 = 0.08 • 360 = 28.8
and yet…More Vocabulary
Arc:
• is part of a circle (part of the circumference)
• denoted by the (
arc symbol)
3 Types of Arcs:
Semicircle:
Half a circle 180º
Minor arc:
smaller than a semicircle
(less than 180º
Major Arc:
greater than a semicircle
(greater than 180º)
Arc Addition Postulate
Adjacent Arcs
• Arcs of the same circle that have exactly one point in
common.
• Measures of adjacent arcs can be added like adjacent
angles
Circles and Arcs
Identify the minor arcs, major arcs, and semicircles in . P with point A
as an endpoint.
Minor arcs are smaller than semicircles.
Two minor arcs in the diagram have point A
as an endpoint, AD and AE.
Major arcs are larger than semicircles.
Two major arcs in the diagram have point A
as an endpoint, ADE and AED.
Two semicircles in the diagram have
point A as an endpoint, ADB and AEB.
Circles and Arcs
Find mXY and mDXM in . C.
mXY = mXD + mDY
Arc Addition Postulate
mXY = m
The measure of a minor arc is the
measure of its corresponding
central angle.
XCD + mDY
mXY = 56 + 40
Substitute.
mXY = 96
Simplify.
mDXM = mDX + mXWM
Arc Addition Postulate
mDXM = 56 + 180
Substitute.
mDXM = 236
Simplify.
Circumference and Arc Length
Circumference is the distance around a circle.
Circles and Arcs
Concentric circles have the same center.
A circular swimming pool with a 16-ft diameter will be enclosed in a
circular fence 4 ft from the pool. What length of fencing material is
needed? Round your answer to the next whole number.
Draw a diagram of the situation.
The pool and the fence are concentric circles.
The diameter of the pool is 16 ft, so the
diameter of the fence is 16 + 4 + 4 = 24 ft.
Use the formula for the circumference of a
circle to find the length of fencing material needed.
C= d
Formula for the circumference of a circle
C = (24)
Substitute.
C 3.14(24)
Use 3.14 to approximate .
C 75.36
Simplify.
About 76 ft of fencing material is needed.
Circles and Arcs
Find the length of ADB in . M in terms of
.
Because mAB = 150,
mADB = 360 – 150 = 210.
length of ADB =
mADB
•2
360
length of ADB =
210
•2
360
r
(18)
length of ADB = 21
The length of ADB is 21
cm.
Arc Addition Postulate
Arc Length Formula
Substitute.