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Areas and Perimeters of Circles
Introduction: Area is a measure of the amount of space contained inside a closed figure. Perimeter is a measure of the distance around a closed figure. In this lesson we will examine these concepts for a circle. In particular, the perimeter of a circle is called the circumference.
 
The Lesson:
 
The area and circumference of a circle involve the radius and/or the diameter of the circle. Sometimes we have information about a figure inside the circle or other segments in the circle which we use to calculate the radius or diameter. First, however, we need formulas for the area and circumference of a circle.
 
To derive a formula for the area of a circle, we will examine a diagram of a (partial) regular polygon with sides of length s and number of sides n. A diagram is shown below.
 
Assume that point O is the center of the regular polygon and r, the distance from the center to a vertex, is called the radius of the polygon. This means r is the radius of a circle and O is the center of a circle into which a regular polygon can be inscribed. The perimeter of the polygon is the sum of the n sides of length s and is given by ns. As the number of sides in the polygon increases, the polygon fits more exactly into the circle (becomes closer to the actual shape and size of the circle). We will derive a formula for the area of the polygon in terms of the radius r and use this to calculate the area of a circle.
 
The Polygon:
 
Triangle AOB is a central triangle. There are n such triangles in this polygon, one for each side.
 
Since there are n such triangles in our polygon:
 
AOB has a measure of
MOB has a measure half as large or
 
Since AB is a side of measure s,
 
MB has a measure of
 
Using basic right triangle trigonometry and our right triangle OMB, we see that
 
and
 
This gives us the area of of our sample triangle AOB as
 
A =
 
Since there are n such triangles in this regular polygon, the area of a regular polygon of n sides and radius r is
 
 A =
 
The Circle:
 
Below we show two tables of values for for two large values of n.
 
For a polygon having 175 sides
 
 
 
For a polygon having 2825 sides
 
 
 
As you can infer from the diagram below, the more sides we use in the polygon, the closer the polygon will get to matching the edge of the circle.
 
 
Combine this with the fact that the value of  p for larger and larger values of n, we can conclude that our formula for the total area of the polygon is
 
A = pr2.
 
 
Similarly, we can use or to determine the formula for the circumference of a circle.
 
 
The perimeter of the polygon is ns where n is the number of sides. It happens that p as n gets larger allowing us to conclude that our formula for the total perimeter of the polygon is
 
2pr.
 
Summary:
 
We now have the following two formulas:
 
  • The area of a circle is given by A = pr2 where r is the length of the radius of the circle.
  • The circumference of a circle is given by C = 2pr where r is the length of the radius of the circle. Notice that the circumference could also be written as C = pd where d is the length of the diameter since d = 2r.
 
Let's Practice:
 
  1. A circle has a diameter of 8 feet. What are the area and circumference of this circle?
 
Notice that we are given a diameter of 8 which means that the radius is 4.
 
The area equals:
 
A = pr2 = p(4)2 = 16p square feet
 
The circumference equals:
 
C = 2pr = 2p(4) = 8p feet
 
It is typical to leave answers in terms of p unless there is a need for a decimal approximation of the answer.
 
  1. A right triangle is inscribed in a circle of radius 6 inches as shown in the diagram. What is the area of the shaded region?
 
 
The strategy is to find the area of the triangle and subtract it from the area of the circle.
 
We note that if a right triangle is inscribed in a circle, the hypotenuse is a diameter of the circle because ACB must intercept an arc of twice its measure or 180º which is half a circle. Therefore the length of AB is 12 because AB is a diameter and the radius is given as 6 inches.
 
The area of the triangle is:
 
Atriangle = (½)bh = (½)(12)(5) = 30 square inches
 
The area of the circle is:
 
Acircle = pr2 = p(6)2 = 36p square inches
 
We have the area of the shaded region as 36p – 30   83.1 square inches.
 
  1. A square is inscribed in a circle of circumference 10p. What is the area of the shaded region? A diagram is given below.
 
 
The strategy is to subtract the area of the square from the area of the circle.
 
Since the circumference of the circle is given as 10p
 
C = pd
C = 10p
 
pd = 10p
d = 10
 
In our diagram, AC is a diameter of the circle. Therefore, AC = 10.
 
We will now use this information along with the Pythagorean Theorem to find the length of a side of the square, which we have labeled x in the diagram.
 

2x2 = 100
x2 = 50
x 7.07
 
Notice that the area of the square equals x2
 
Asquare = 50
 
Since r = 5 because the diameter is 10, the area of the circle is
 
Acircle = pr2 = p(5)2 = 25p
 
The area of the shaded region is
 
25p – 50   28.5

Examples
Example A circle has an area of 40p. What are the radius and the circumference of this circle?
What is your answer?
 
Example A right triangle having an area of 16 and an altitude of 3 is inscribed in a circle . What are the area and circumference of this circle?
What is your answer?
 



M Ransom

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