I like pie! Do you
like pie? What is
the shape of pie?
Circles
Using that special number!
Circles
A bicycle odometer uses a magnet
attached to the wheel and a sensor
attached to the bicycle frame. Each
time the magnet passes the sensor,
the odometer registers the distance
traveled. This distance is the
circumference of the wheel.
Here is an example of the odometer on my
bicycle. The pictures show the odometer
reading and the sensor on the wheel.
Circle:
The set of all points in a plane that
are equal distance from the center. (a circle
is named by it’s center)
Radius:
a line segment with one endpoint
at the center of the circle and the other
endpoint on the circle. Radii is pleural of
radius.
Diameter:
a line segment with endpoints
on the circle and it passes through the center
of the circle.
Chord:
a line segment with its endpoints on
a circle.
Do you like pie?
Describe the shape of a pie.
This will help you remember an important
number called pi. This special number will be
used to solve problems involving circles.
Pi: (π) the ratio of the
circumference of a circle to it’s
diameter. Pi is an irrational
number that is approximated by
the rational numbers π ≈ 3.14 or
22/7.
π ≈
3.14159265358979323846…,
although Pi has been computed to
more than a trillion places, for
circumferences and area using
3.14 is usually enough.
Pi can be found to over a trillion places using
computers today. Below is a fraction of what
the number looks like when finding Pi.
Circumference:
the distance around a
circle.
Like perimeter, the
circumference is the distance
around the outside of a figure
(circle). Unlike perimeter, in
a circle there are no straight
segments to measure, so a
special formula is needed.
C = πd when you know the diameter
C = 2πr when you know the radius
Area of a circle: the number of square
units needed to cover the surface of a figure.
Again a special formula is needed because
there are no straight segments to measure.
A = πr2
Find the area of the circle:
when working with circles, be sure you are
using the radius, i.e., A = πr². In this diagram,
10 is the diameter. The radius is half of the
diameter.
Example:
Here is an example of a problem
you will be asked to solve using circle formulas.
In this figure you are asked to find the area
of the shaded portion.
There are many applications used everyday
using formulas for circles. Take the example
below.
A Ferris wheel has a
diameter of 56 feet and
makes 15 revolutions per
ride. How far would
someone travel during a
ride?
C = πd
C = πd
C = 175.9291886…
175.9291886 · 15 =
2638.937829 feet
Solving word problems
Many car tire manufactures guarantee their
tires for 50,000 miles. If the average tire
has a 2 ft diameter, how many revolutions does
the manufacturer guarantee?
1 revolution
C = πd
C = 3.14 · 2
C = 6.28ft
Guaranteed
mileage
Feet/mile
5280/6.28 · 50,000 ≈
42,038,216.56
revolutions
Solving word problems
Graph a circle with center (3, 1) that passes
through (3, -1). Find the area and
circumference, both in terms of π and to the
nearest tenth. Use 3.14 for π
A = πr2 C = πd
A = π · 22 C = π · 4
A = 4π C = 4π
A = 4 · 3.14 C = 4 · 3.14
A=12.56 units2 C=12.56units
Area & Circumference
of Circles
Remember to use that special
number called pi
1) Find the area of the circle.
nearest tenth)
a)
b)
c)
d)
314.2 units
78.5 units
212 units
31.4 units
(round to the
2) Find the circumference and area of a circle
with a diameter of 15 cm.
a)
b)
c)
d)
A = 126.9 cm2 C = 40.4 cm
A = 176.6cm2 C = 47.1 cm
A = 452.2 cm2 C = 144 cm
A = 153.9 cm2 C = 44.0 cm
3) Find the circumference
and area the circle.
a)
b)
c)
d)
A = 6.2m2, C = 1.9m
A = 7.1m2, C = 2.3m
A = 4.5m2, C = 7.5m
A = 50.2m2, C = 25.1m
4) Find the number of
square inches in the
area of the shaded
region of this square.
a)
b)
c)
d)
95.0 sq. in.
285.1 sq. in.
25.9 sq. in
78.4 sq. in
5) Find the area of this figure.
(round to the nearest square inch)
a)
b)
c)
d)
356.5 sq. in.
457.1 sq. in.
658.1 sq. in.
1060.2 sq. in.
6) Find the area of this
figure. (round to the nearest square
cm)
a)
b)
c)
d)
56.5 sq. cm
140.5 sq. cm
310.2 sq. cm
478.2 sq. cm
7) Butch, the dog, is leashed to
the corner of the house when
he is outdoors alone. The leash
is 20 feet long. Find the
amount of ground area available
for Butch to run around in.
(assume the corner of the house to be a
right angle).
a)
b)
c)
d)
251.3 sq. ft.
314.2 sq. ft.
628.3 sq. ft.
942.5 sq. ft.
8) Find the area of the shaded portion of the
circle. (round to the nearest tenth of a meter)
a)
b)
c)
d)
49.0 m2
153.9 m2
197.3 m2
297.7 m2
9) Find the radius of a circle with an area of
169 πin2.
a)
b)
c)
d)
26 in.
13 in.
9 in.
4 in.
10) Graph a circle with center (3, -1) that
passes through (0, -1). Find the area and
circumference, to the nearest tenth.
a)
b)
c)
d)
A = 176.6 units2, C = 56.3 units
A = 153.9 units2, C = 44.0 units
A = 28.3 units2, C = 18.8 units
A = 113.0 units2, C = 37.7 units
11) If the diameter of
the average automobile
tire is 2 ft, about how
many revolutions does
the wheel make for
every mile driven? (Hint:
1 mi = 5280 ft.)
a)
b)
c)
d)
720
640
840
632