How Much Does the Area Increase When the Diameter of a Circle Increases by 40%?

Understanding the relationship between the diameter and area of a circle is fundamental in various fields, including geometry, physics, and engineering. This article aims to explore the exact percentage increase in the area of a circle when the diameter is increased by 40%. By the end of this article, you'll be able to calculate the exact percentage increase with ease.

Understanding the Basic Formulae

The area ( A ) of a circle is determined by its radius ( r ) using the formula ( A pi r^2 ). The diameter ( D ) of a circle is twice the radius, i.e., ( D 2r ).

Capturing the Original and New Diameter and Radius

Let's denote the original diameter as ( D ), and the original radius as ( r frac{D}{2} ).

Calculating the Original Area

The area of the circle is given by:

[ A pi r^2 pi left(frac{D}{2}right)^2 frac{pi D^2}{4} ]

Increasing the Diameter by 40%

When the diameter is increased by 40%, the new diameter ( D' ) becomes:

[ D' D 0.4D 1.4D ]

The new radius ( r' ) is then:

[ r' frac{D'}{2} frac{1.4D}{2} 0.7D ]

Calculating the New Area

The new area ( A' ) with the new radius is:

[ A' pi (r')^2 pi (0.7D)^2 pi cdot 0.49D^2 frac{49pi D^2}{100} ]

Calculating the Increase in Area

The increase in area ( Delta A ) is given by the difference between the new area and the original area:

[ Delta A A' - A frac{49pi D^2}{100} - frac{pi D^2}{4} ]

Converting ( frac{pi D^2}{4} ) to have a denominator of 100:

[ frac{pi D^2}{4} frac{25pi D^2}{100} ]

Thus, the increase in area is:

[ Delta A frac{49pi D^2}{100} - frac{25pi D^2}{100} frac{24pi D^2}{100} ]

Calculating the Percentage Increase

The percentage increase in area is calculated as:

[ text{Percentage Increase} left(frac{Delta A}{A}right) times 100 left(frac{frac{24pi D^2}{100}}{frac{pi D^2}{4}}right) times 100 ]

Simplifying this:

[ left(frac{24}{100} times frac{4}{pi D^2} times frac{100}{pi D^2}right) times 100 frac{24 times 4}{25} 96% ]

Example with Radius

Let's consider a circle with a radius of ( x ) cm. The original area is ( pi x^2 ). When the radius increases by 47%, the new radius is ( 1.47x ) cm. The new area is:

[ text{New Area} pi (1.47x)^2 pi cdot 2.1609x^2 2.1609pi x^2 ]

The increase in area is:

[ Delta A 2.1609pi x^2 - pi x^2 1.1609pi x^2 ]

The percentage increase is:

[ text{Percentage Increase} left(frac{Delta A}{pi x^2}right) times 100 frac{1.1609pi x^2}{pi x^2} times 100 116.09% ]

Using a Unit Radius

Let ( r 1 ) unit. The original area is ( pi (1)^2 pi ). When the radius increases to ( 1.47 ) units, the area becomes:

[ A_2 pi (1.47)^2 pi cdot 2.1609 2.1609pi ]

The increase in area is:

[ A_2 - A_1 2.1609pi - pi 1.1609pi ]

The percentage increase is:

[ text{Percentage Increase} left(frac{1.1609pi}{pi}right) times 100 116.09% ]

Conclusion

As shown through both numerical and theoretical examples, when the diameter of a circle increases by 40%, the area of the circle increases by 96%. For a more dramatic increase to 116.09%, a 47% increase in the radius is necessary.