List of Formulas for the Circumference of a Circle
Depending on the given information, three formulas are used to calculate the area of a circle. The formulas for the area make use of the irrational number Pi, denoted by the Greek letter . Pi is a mathematical constant and is the ratio of a circle’s circumference to its diameter. The value of pi is 3.14 or \(\frac{22}{7}\).
Now that we know about the nature and value of \(\pi\), next, we look at the list of formulas that are related to calculating the area of a circle.
1. Area of a circle, given the radius = \(\pi r^2\)
2. Area of a circle, given the diameter = \(\frac{\pi}{4}d^2\)
3. Area of a circle, given the circumference = \(\frac{C^2}{4\pi}\)
Area of a circle Formula
To calculate the area of a circle, you either require the radius, the diameter or the circumference of the circle.
To find a circle’s area, multiply the square of radius by (pi).
We know that the diameter is 2 times the length of the radius, hence, the second formula for the area of a circle is the product of \(\frac{\pi}{4}\) and the square of the diameter.
The formula for calculating the area of a circle in terms of the circumference is \(\frac{\text{Circumference}^2}{4\pi}\).
Mathematically, the area of a circle can be expressed as:
Area (A) of a circle = \(\pi r^2\).
Area (A) of a circle = \(\frac{\pi}{4}d^2\).
Area (A) of a circle = \(\frac{C^2}{4\pi}\).
Where,
- r is the radius of the circle.
- d is the diameter of the circle.
- C is the circumference of the circle.
Therefore, we can use the formula, \(\pi r^2\) or \(\frac{\pi}{4}d^2\) or \(\frac{C^2}{4\pi}\) , to find the area of a circle.
Solved Examples
Example 1: Find the area of a circle with a radius of 14 cm. Use \(\frac{22}{7}\) for \(\pi\).
Solution:
\(A=\pi r^2\) Write the formula for area given the radius
\(~~~=\frac{22}{7}\times 14^2\) Substitute \(\frac{22}{7}\) for and \( 14^2 \) for radius
\(~~~=\frac{22}{7}\times 196\) Evaluate \( 14^2 \) and divide out the common factor.
\( ~~~= 616 \) Multiply
The area is 616 square centimeters.
Example 2: Find the area of a circle with diameter of 7 cm. Use \(\frac{22}{7}\) for π.
Solution:
\(A=\frac{\pi}{4}d^2\) Write the formula for area of a circle given the diameter
\(~~~=\frac{22}{7\times 4} \times 7^2\) Substitute \(\frac{22}{7}\) for and 7 for diameter
\(~~~=\frac{22}{7\times 4} \times 49\) Evaluate \(7^2\) and divide out the common factor
\( ~~~= 38.5 \) Simplify
The area is 38.5 square centimeters.
Example 3: Find the area of a circle with a circumference of 11 cm. Use \(\frac{22}{7}\) for π.
Solution:
\(A=\frac{C^2}{4\pi}\) Write the formula for area of a circle given the circumference
\(~~~~=\frac{11^2}{4\times \frac{22}{7}}\) Substitute \(\frac{22}{7}\) for and 11 for circumference
\(~~~~=\frac{11\times 11 \times 7}{4\times 22}\) Evaluate \(11^2\) and cancel out the common factor
\( ~~~~= 9.625 \) Simplify
The area is 9.625 square centimeters.
Example 4: Find the diameter of a circle with an area of 154 square units.
Solution:
The area of the circle is 154 square units.
\(A=\pi r^2\) Write the formula for area of a circle given the radius
\(154=\frac{22}{7}\times r^2\) Substitute \(\frac{22}{7}\) for π and 154 for area
\(\frac{154\times 7}{22}=r^2\) Transposing all terms to one side and the unknown to one side
\(r^2= 49 \) Simplify
\( r = 7 \)
Therefore,
\(d = 2\times 7 \)
\( ~~~= 14 \) units
Hence, the diameter of the circle with an area of 154 square units is 14 units.
Example 5: On Friday night, John and his friends ordered pizza. The side length of each slice was 10 cm. Calculate the area of the pizza that John ordered. You can assume that the side length of a pizza slice equals the radius of the pizza.
Solution:
\(A=\pi r^2\) Write the formula for area.
\(~~~=3.14\times 10^2\) Substitute 3.14 for and 10 for radius.
\(~~= 3.14\times 100 \) Evaluate \(10^2\).
\( ~~~= 314 \) Multiply.
The area of the pizza is 314 square centimeters.
In two-dimensional space, the area of a circle is the region occupied by the circle.
The circumference of a circle is equal to twice the product of pi(π) and the radius of the circle, that is 2πr.
Pi or or “π” as a symbol, is an irrational number and its value is taken to be “3.143” in its decimal form in calculations.Its exact value can never be determined as irrational numbers are non terminating and non recurring decimals. When this decimal of value of is written as a fraction, the value of is taken to be “22/7” for the purpose of calculations.