Circles
Intro
Have you ever placed your fingertip on still water? The tip of your finger produces
ripples in the water which seem to form perfect circles. When riding your bike on an
even surface, have you ever wondered why your wheels give a smooth ride?
Well, it is because circles have a very special property—every point on the circle is the
same distance from its center.
A circle with center P is often referred to as circle P or P (“” is the symbol for circle).
When two or more circles share the same center, they are called concentric circles.
Parts of A Circle
Below you will find a diagram of circle P which illustrates the parts of a circle. It is
important to familiarize yourself with these parts since each has its own unique
properties.
Secant
Diameter

P
Tangent
Radius – A segment which joins the center of a circle
to a point on the circle.
Diameter – A chord which passes through the center
of a circle.
Radius
Chord
Chord – A segment which joins to points on a circle.
Tangent – A line that intersects a circle at exactly one
point.
Secant – A line that intersects a circle at two points.
4 Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals, and circles in solving problems.
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Pi
The ratio of the circumference or a circle, C, to its diameter, d, is a
number called pi, or π (Greek symbol).

C
, where C is the circumference and d is the diameter
d
of a circle.
π is a an irrational number; commonly used approximations for π
are 3.14 and
22
.
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Figure 1: Image under Public Domain
http://en.wikipedia.org/wiki/Image:Pi_eq_C_over_d.svg
Oftentimes, the problem will give you an idea of which approximation to use. For example, if the
problem asks for an exact answer, we write our solution using π:
Question: If the radius of a circle is 5, find its circumference. Give an exact answer.
Solution: The circumference of a circle is 2 πr. Therefore, 2 π(5) = 10 π. Since the question asked
for the answer to be in exact form, we leave it as is, 10 π.
If the problem asks for an approximation:
Question: If the radius of a circle is 5, find its circumference. Give an approximation.
Solution: The circumference of a circle is 2 πr. Therefore, 2 π(5) = 10 π. However, π can be
approximated to be about 3.14. So, we would take this a step further:
10(3.14) ≈ 31.14.
Unless the problem states to approximate your answer, you can leave your answer in terms of π
(meaning you do not have to approximate π).
Circumference
Carpenters oftentimes use make-shift tools to measure and/or
construct 90° corners, level lines and circles or arcs. When
creating a circle, the sometimes take a piece of string with a
pencil tied to one end and a thumbtack at the other. They place
the thumbtack at the center of their circle, hold the string taught
and sweep out a circle with their pencil.
The pencil draws the perimeter of the circle—the circumference
is the distance around it.
Figure 2: Image under public domain.
http://www.instructables.com/id/How-to-draw-a-circlewith-string/
4 Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals, and circles in solving problems.
2
The circumference of a circle is given by the formula: 2πr, where r is the radius of the circle.
For example, suppose you wanted to find the circumference of a circle with a radius of 4cm.
Using our formula, 2πr, we can substitute for our radius: 2π(4) = 8π. Therefore, the circumference
of a circle with a radius of 4cm is 8πcm.
Area of a Circle
Circumference
The area of a circle is the region bounded by the circumference.
The area of a circle is given by the formula: πr2, where r is the radius
of the circle.
Area
For example, suppose you wanted to find the area of a circle with
radius of 4cm.
Using our formula, πr2, we can substitute for our radius: π(42) = 16π. So, the area of a circle with a
radius of 4cm is 16πcm2.
Check these out!
Here are a few types of examples you may encounter which use the formula for the area of a circle
in a deeper manner.
Example 1
The area of a circle is 25πcm2. Find the diameter of the circle.
Solution
Since we are given the area of a circle, we know to work with the formula, πr2.
πr2= 25π Since we are told the area is 25π we set that equal to the area of a circle.
r 2 25



We want to isolate r (radius); to do so, divide each side by π.
r2 = 25
r 2  25
r=5
In order to isolate r, we must take the square root of each side.
Our radius is 5.
Since the diameter is double the length of the radius, the diameter of the circle is 10cm.
4 Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals, and circles in solving problems.
3
Example 2
Find the area of the shaded region.
Solution
8in.
In order to find the area of the shaded region, we will use the formula
for the area of a circle, πr2.
3in.
Notice we have two concentric circles—the smaller has a radius of 3
inches, the larger a radius of 7 inches. In order to find the area of the shaded region, we can
simply subtract the area of the smaller circle from the area of the larger circle.
Area for small circle: π(32) = 9π in2.
Area for larger circle: π(72) = 49π in2.
Subtract smaller from larger: 49π - 9π = 40π in2.
Note: You can only add or subtract “π” as long as both numbers contain it. In other words, if you try to add
3π + 6, the answer will be 3π + 6. However, if you add 3π + 6π, the answer will be 9π because they both
contain “π”.
Example 3
Find the area of the shaded region.
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Solution
Since we are finding the area of a part of a circle, we will use the
formula for the area of a circle, πr2.
1
4
πr2 = π(62) = 36π (area of entire circle). However, the shaded region is of the entire circle; so
taking
1
(36π) = 9π.
4
Therefore, the area of the shaded region is 9π.
4 Knowledge of Geometry
Use properties of lines (i.e., slope and midpoint), angles, triangles, quadrilaterals, and circles in solving problems.
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