Circle Handout
CIRCUMFERENCE AND AREA OF CIRCLES
When working with circles, we need to be aware of a couple of vocabulary words that are
unique to circles.
RADIUS & DIAMETER
RADIUS refers to the distance from the center of a circle to its edge. Its plural is "radii."
DIAMETER refers to the distance from one edge of a circle to the other edge if it is
measured THROUGH THE CENTER of the circle.
diameter
radius
Notice that if we put together TWO radii, we would end up with a diameter. Because of
this, we can say that
2  radius  diameter .
PI
The symbol  (read "pi"), refers to a very special number. The number is an "irrational"
number, which means its decimal portion never ends and never repeats.  is a number,
just as 5 or 73 is a number. It's first ten digits are 3.141592653.
If we want to actually work with  , we will generally use an approximation of  : either
3.14 as a decimal or 227 as a fraction. Remember, these numbers are not exactly  , but
are about equal to  . The approximation 3.14 is used the majority of the time.
If I want to do the problem 2   , the EXACT answer would be, simply, 2  . An
APPROXIMATE answer would be to use either approximation for  : Using 3.14, we
would have 2  3.14  6.28 . If we use 227 we have 12  227  447 , or 6 72 .
CIRCUMFERENCE
CIRCUMFERENCE refers to the distance around the outside of a circle. It is similar to
the idea of "perimeter" for polygons. Its units are always just "straight" units (inches,
meters, etc). To find the circumference of a circle, we use the formula
C  d 
where d represents the diameter of the circle. Since we know that the diameter is the
same as twice the radius, the formula for circumference can also be written
C  2  r 
where r represents the radius of the circle.
EXAMPLE: Find the circumference of the circle shown. Give an exact answer, and then
an approximation, using 3.14 for  .
7 inches
Notice that the radius is given as 7 inches, so we use C  2  r   . This gives us:
C  2  7    14 inches
for our EXACT answer. Notice that the  symbol appears in the exact answer: this will
always be the case. For an approximation, we will use 3.14 for  . That give us:
C  14    14  3.14  43.96 inches.
The use of the  symbol is our way of showing the  doesn't EXACTLY equal 3.14, but
it is APPROXIMATELY equal to 3.14.
So, the final answers would be: The exact circumference is 14 inches, and the
approximate circumference is 43.96 inches.
AREA
AREA is the same for circles as it is for polygons: it refers to the surface that a circle
covers, and is always given as a measure of SQUARE units (square miles, square inches,
etc). To find the area of a circle, we use the formula
A   r2
We do not have an equivalent formula using the diameter, but if the diameter of a circle is
given, we can always divide it by 2 to come up with the radius, and then use the formula
above. The same rules apply to exact and approximate answers for Area as they do for
Circumference.
EXAMPLE: Find the area of the circle in the example above. Give an exact AND an
approximate answer, using 3.14 for  .
The radius was given to us as 7 inches, so we plug that into our formula A    r 2 :
A    7 2    49  49 square inches.
This is our exact answer. We will use 3.14 for  to find our approximation:
A  49  49  3.14  153.86 square inches.
So our final answer is: The area is 49 square inches exactly, and approximately 153.86
square inches.
**NOTE: In your homework, #41 in section 8.3 and #25 in section 8.4 specify using
for  . The answers in the back of the book SHOULD read:
3
1
41. 220
25. 792
7 ft. or 31 7 ft.
7 sq. inches or 113 7 sq. inches.
This is in keeping with the principle that says, “If a problem gives you a fraction, give
your answer in fraction form.”
22
7
PROBLEMS TO PRACTICE:
For the problems below, find the circumference and the area of the given circles. Math
75 students: do circumference after §4.4 and review after §8.3; do area after §8.4. Give
both an exact answer and an approximate answer for each (use 3.14 as an approximation
for  ). Be sure to include your labels. Answers are printed on the last page: Be sure to
ask an instructor if you have any questions.
1.
Circumference: Exact: __________
10 cm
Approx: ___________
Area: Exact:___________
Approx: ________
2.
Circumference: Exact: __________
6 feet
Approx: ___________
Area: Exact:___________
Approx: ________
3.
Circumference: Exact: __________
Approx: ___________
8 miles
Area: Exact:___________
Approx: ________
4.
Circumference: Exact: __________
Approx: ___________
1 inch
Area: Exact:___________
Approx: ________
Answers:
1. Circumference: exact = 20  cm. approximate = 62.8 cm.
Area: exact = 100  sq. cm approximate = 314 sq. cm.
2. Circumference: exact = 6  ft. approximate = 18.84 ft.
Area: exact = 9  sq. ft. approximate = 28.26 sq. ft.
3. Circumference: exact = 8  mi. approximate = 25.12 mi.
Area: exact = 16  sq. mi. approximate = 50.24 sq. mi.
4. Circumference: exact = 2  in. approximate = 6.28 in
Area: exact =  sq. in. approximate = 3.14 sq. in.