Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
p.1
Geometry
Lesson Plans
Sections 7-6: Circles: Circumference and Arc Length
4/09/12
Warm-up (20 mins)
In groups of 2 or 3, pick up a tangram puzzle and a ruler.
 Put the pieces together to solve the puzzle
 Find the total area of the figure: Write down each shape and its area.
Please be careful with the pieces – they have to be reused in each class!
Have a couple of groups volunteer to show their results. (5 mins)
Homework Review (10 mins) – ask for any questions on homework (from previous week for 4B)
and area worksheet (do number 34 for 4B at least)
Homework (H)
p. 376 – 377, #1-20, 22, 29-31, 34-36
Homework (R)
p. 376 – 377, #1-20, 22, 29, 31, 36
Statement of Objectives (5 mins)
The student will be able to find
 measures of central angles and
 circumference and arc length.
Teacher Input (50 mins)
Vocabulary
 A circle is the set of all points equidistant from a given point called the center of the circle.
 A radius is the line segment from the center to a point on the circle. In circle P below, PC , PA , and
PB are radii.
 Congruent circles have congruent radii.
 A diameter is a line segment that contains the center and has both endpoints on the circle. In circle
P below, AB is a diameter.

A central angle is an angle whose vertex is the center of the circle. In circle P below, ∠APC is a
central angle whose measure is 510
Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
p.2
Arcs:
 Semicircle – half a circle. arc AB is a semicircle. mAB = 180

Minor arc – less than half a circle. arc AC is a minor arc. mAC = m∠APC = 51.

Major arc – more than half a circle. arc CAB is a major arc. mCAB = 360 - m∠CPB = 360 – (180 –
51) = 129.
Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measures
of the two arcs. mAC + mCB = mACB
Example 1: Identify the following in circle P:
Elizabeth Pawelka



Circles: Circumference and Arc Length
4/09/12
Minor arcs:
o AC, DB, CB, AD
Major arcs:
o ACD, CBA, BDC, DAB
Semicircles:
o ACB, BDA, CBD, DAC
Example 2: Find the measures of the central angles and arcs in this pie chart:
p.3
Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
p.4
Why People Think Teens
Cause the Most Accidents
8%
16%
Reckless
20%
Inexperience
Want to have fun
Don't care
32%
Too young
Speed
16%
8%






Reckless: 16% of 360 = 57.6
Inexperience: 32% of 360 = 115.2
Want to have fun: 8% of 360 = 28.8
Don’t care: 16% of 360 = 57.6
Too young: 20% of 360 = 72
Speed: 8% of 360 = 28.8

Circumference is the distance around a circle.

pi (π) is the ratio of the circumference of a circle to its diameter: π =

diameter = 2*radius => C = 2πr

π ≈ 3.14159… Note that this is approximate because π is non-terminating.
Example 3: Find the exact circumference of these circles:
C
or C = π*d
d
Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
p.5
d = 16 cm
C = 16π
r = 5.3 cm
C = 10.6π cm
C = 12π
C = 8π
Example 4: What is the radius of a circle with circumference = 18πm? (9m)

Concentric circles – two circles that lie in the same plane and have the same center
Example 5: Turning Radius of a car
If a car has turning radius of 16.1 ft and the distance between the front tires is 4.7 ft, how much farther
does the outside tire travel?
Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
p.6
Circumference of inner circle: radius = 16.1 – 4.7 = 11.4, so circumference = 2*11.4*π = 22.8π
Circumference of outer circle = 2*16.1π = 32.2π
Difference = 32.2 π - 22.8 π = 9.4 π ≈ 29.531 ft.
Example 6: The radii of 2 concentric circles are 4.5cm and 5.2cm. How much larger is the
circumference of the larger circle than the smaller circle? (1.47cm)
Example 7:
Elizabeth Pawelka
Circles: Circumference and Arc Length
4/09/12
Arc Length: ratio of the measure of the arc and the circumference of a circle:
360
Length of arc AB =
mAB
* 2πr
360
Example 8: Find length of arc CE:
p.7
Elizabeth Pawelka
Circles: Circumference and Arc Length
C = 16π; length of arc CE =
90
* 16π = 4π
360
Example 9: Find length of arc HI:
C = 16π; length of arc CE =
60
* 12π = 2π
360
4/09/12
p.8
Elizabeth Pawelka
Circles: Circumference and Arc Length
Example 10: Find length of arc HEY:
∠HOY = 120 and 240. Length arc HEY =
240
*30π = 20 π in
360
4/09/12
p.9
Elizabeth Pawelka
Example 11:
Circles: Circumference and Arc Length
4/09/12
p.10
Elizabeth Pawelka
Circles: Circumference and Arc Length
Example 12:
Closure (5 mins)
Today you learned to find
 measures of central angles and
 circumference and arc length.
Tomorrow you will learn to find the area of circles and sectors
Homework (H)
p. 390 # 9 – 39, 42 – 47, 52, 53, 59
Homework (R)
p. 390 # 9 – 39, 42 – 47, 53
4/09/12
p.11