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  .
Notice that the radius is given as 7 inches, so we use C  2  r   . This gives us:
7 inches
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.
PROBLEMS TO PRACTICE:
For the problems below, find the circumference and the area of the given circles. 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.