Area of a Circle
The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter.
we use the Greek letter Pi (pronounced Pi) to represent the ratio of the circumference of a circle to the diameter.
In the last lesson, we learned that the formula for circumference of a circle is: C equals Pi times d. For simplicity, we use Pi = 3.14.
We know from the last lesson that the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula:
The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle.
Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2 However, it is easier to use one of the following formulas:
A = Pi times r squared or A = Pi times r times r
where A is the area, and r is the radius. Let's look at some examples involving the area of a circle.
In each of the three examples below, we will use Pi= 3.14 in our calculations.
Example 1: The radius of a circle is 3 inches. What is the area?
Solution: A = Pi times r times r
A = 3.14 · (3 in) · (3 in)
A = 3.14 · (9 in2)
A = 28.26 in2
Example 2: The diameter of a circle is 8 centimeters. What is the area?
Solution:
8 cm = 2 · r
8 cm ÷ 2 = r
r = 4 cm
A = Pi times r times r
A = 3.14 · (4 cm) · (4 cm)
A = 50.24 cm2
Example 3: The area of a circle is 78.5 square meters. What is the radius?
Solution: A = Pi times r times r
78.5 m2 = 3.14 · r · r
78.5 m2 ÷ 3.14 = r · r
25 m2 = r · r
r = 5 m
Summary Given the radius or diameter of a circle, we can find its area. We can also find the radius (and diameter) of a circle given its area.
The formulas for the diameter and area of a circle are listed below:
A = Pi times r squared or A = Pi times r times r
Properties of a Circle
The circumference of a circle is the limit of the perimeters of the inscribed regular polygons.
Please see Figure 1 below. The ratio of circumference to diameter is always constants, denoted by π, for a circle with the radius r as the size of the circle is change. Let d be defined as diameter and c as circumference. Then, as observe, since d=2r, C=2πr the ratio of c and d is C÷D=2πr÷2r=π.
For example, Let
R=3.14 cm
D=7.09 cm
Then, (circumference) = (r.d) =22.26 cm
(Circumference)÷d= (r.d) =22.26 cm÷7.09=3.14 cm or π=3.14
Figure 1. The ratio of
Circumference
of diameter.
Circumference
of diameter.
Traditional or partitioning method
Figure 2. Traditional or partitioning method
In determining the equal parts of a circle into three by using the traditional or partitioning method, there are three ways of showing it:
1. Find angle α using protractor.
2. Find arc length β using a piece of string around the circumference of a circle.
3. Find the area of the sector of a circle by using the formula for the sector of a circle.
First, somebody who would actually take a restaurant would draw laughs from others but would be able to divide the circle into the three equal parts. See the Figure above. Each of which is=C÷3=360◦=120◦ where α is the measure of the central angle of the shaded part.
Second, wrapping a piece of string around a circumference of a circle would also help. Dividing the string into three parts would let us mark the circumference in thirds that is β=C÷3=2πr÷3 where β is the measure of the circumference or the yellow shaded part.
Lastly,given the angle α, we can use the formula for the sector of a circle. The angle of each sector is α=120◦ .Since the area of the sector is proportional to the angle.
A =α÷360◦ (π) (r2)
= (πr2) (α) ÷360◦
= (π (1)) (120◦) ÷360◦
=1÷3π or 1.047
Thus, we have shown that either by using a protractor or by getting a string around the circumference of a circle or by using the formula for the sector of a circle, we would really determine that each part or the sector shares the same area which is 1÷3 or 1.047.
