Derive and use formulas to find the area and circumference of circles

1. Area and Circumference
The student is able to (I can):
• Develop and use formulas to find the area and
circumference of circles

2. The irrational number  is defined as the ratio of the
circumference of a circle to its diameter. This means that if
we know the diameter, we can find the circumference by
reversing this:
Since we know that the diameter is twice the radius, we can
rewrite the formula if we are given the radius:
C
d
C d
 
 
2C r 

3. We can take any parallelogram and make a rectangle out of
it:
parallelogram formula – the area formula of a parallelogram
is the same as the rectangle: A = bh
(Note: The main difference between these formulas is that
for a rectangle, the height is the same as the length of a side;
a parallelogram’s side is not necessarily the same as its
height.)

5. If you cut a circle into wedges, you can arrange the
wedges into a parallelogram-shaped figure:
radius
1
2
circumference
A = bh
1
circumference radius
2
A
 
1
2 radius radius
2
A 
2
A r 

6. Depending on the circumstances, you may either be asked to
approximate the circumference/area or you may be asked
for the exact circumference/area. What’s the difference?
•  is an irrational number, which means that it keeps
going and never repeats. We can approximate it as
3.14159…, but if we want an exact number, we just use
the symbol, , which represents the entire value of .
• Sometimes we want an approximation, especially for
practical purposes such as measuring things. Other
times, we want the exact value, especially if we’re going
to be using the value in further calculations.
• If you are asked for an exact answer, don’t substitute
anything in for , just leave it as part of your answer.

7. Examples
1. Find the exact area of a circle whose diameter is 16 in.
2. Find the diameter and area of a circle whose
circumference is 22 cm.
3. Find the radius of a circle whose area is 81 sq. ft.

8. Examples
1. Find the exact area of a circle whose diameter is 16 in.
r = ½(16) = 8 in. A = r2 = (82) = 64 in2
2. Find the diameter and area of a circle whose
circumference is 22 cm.
22 = d
d = 22 cm
A = (112) = 121 cm2
3. Find the radius of a circle whose area is 81 sq. ft.
81 = r2
81 = r2
r = 9 ft