How Was The Area Formula for a Circle Derived?

You might be knowing what is the area of a circle. While solving mensuration problems, you have used following formulae.  

• Rectangle: Perimeter = $2 \left(l + b \right)$ and Area = $lb$, where $l$ and $b$ are length and breadth of a rectangle.
• Circle: Perimeter (Circumference) = $2 \pi r$ and Area = $ \pi r^{2}$, where $r$ is the radius of a circle.

Also, you might be knowing how you can derive the formulae for perimeter and area of a rectangle. 

But do you know how the area formula for a circle is derived?

What is the Area of a Circle and How is it Derived?

You might be aware that area or space occupied by a rectangle is equal to the product of its length and breadth, i.e.,  Area = $lb$.

You can use the formula for area of a rectangle to find the area of a circle also.

To understand this let’s consider a circle of radius $r$ and divide into small sectors as shown in the figure below:

The perimeter (circumference) or the length of the boundary of a circle is $2 \pi r$.

Note: Circumference of a circle is equal to sum of the lengths of the arcs = $ \pi r + \pi r = 2 \pi r $.

The circle now can be cut-up to form a rectangle. When the central angle of each sector becomes very small or nearly diminishes, the curves on the bottom and top of the rectangle straighten out to form a straight line with length PI r units while the width of the rectangle is r units.

When you consider a very small portion of the circumference of a circle, it is approximately a straight line. For that reason only, you see the surface of earth as flat (although it is curved),  because you are seeing a very small portion of the Earth surface while on the ground.

Thus the area of the circle approximates the area of the rectangle.

And now you can find the area of rectangle obtained as Area = $ \pi r \times r = \pi r^{2}$

Let’s Code With Python

Find the circumference and area of a circle for a given radius.

#Circumference & Area of a Circle
PI = 22/7 #Define PI
#Ask for radius

r = float(input('Enter radius '))

C = 2 * PI * r #Calculate Circumference
A = PI * (r ** 2) # Calculate Area

#Print results
print('Circumference of circle of radius ', "{:.2f}".format(r), 'is', "{:.2f}".format(C), 'unit')
print('Area of circle of radius ', "{:.2f}".format(r), 'is', "{:.2f}".format(A), 'sq unit')

Conclusion

Rectangle and triangle are two basic plane figures in geometry and one can use these two figures as base to find the perimeter and area of other plane figures.

Practice Problems

(Take $ \pi = \frac {22}{7}$)

1. For a circle of radius $2.8 cm$, find
   1. Circumference
   2. Area 
2. For what length of radius are the area and circumference of a circle numerically equal?
3. If the circumference of a circle is $88 inch$, then find the area of a circle.
4. If the area of a circle is $0.385 m^{2}$, then find the circumference of a circle.
5. What will the percentage change in the area of a circle if its circumference increases by $10 \%$?
6. What will the percentage change in the circumference of a circle if its area decreases by $10 \%$?

Recommended Reading

• Understanding Successive Discount
• Difference Between Infinity and Not Defined
• Difference Between Exponent and Power

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