Understanding the Impact of Radius Increase on a Circle's Area

When the radius of a circle is increased, the area of the circle also increases. This relationship is governed by the formula for the area of a circle, which is given by ( A pi r^2 ). By increasing the radius by a certain percentage, we can mathematically determine the new area and understand the extent of this increase.

What is the Area of a Circle if its Radius is Increased by 20?

Let's consider a circle with an initial radius of 1 unit. Using the formula ( A pi r^2 ), the original area is:

( A_{original} pi (1)^2 pi )

Now, let's increase the radius by 20%. The new radius is 1.2 units. Using the area formula again, the new area is:

( A_{new} pi (1.2)^2 pi times 1.44 1.44 pi )

The increase in area can be calculated by the ratio of the new area to the original area:

( frac{A_{new}}{A_{original}} frac{1.44 pi}{pi} 1.44 )

This means the area has increased by 44%:

( (1.44 - 1) times 100% 44% )

Mathematical Derivation

Given a circle with radius ( r ), the original area is:

( A_{original} pi r^2 )

When the radius is increased by 20%, the new radius is ( 1.2r ). The new area is:

( A_{new} pi (1.2r)^2 pi times 1.44r^2 1.44 pi r^2 )

Dividing the new area by the original area gives:

( frac{A_{new}}{A_{original}} frac{1.44 pi r^2}{pi r^2} 1.44 )

This confirms that the area has increased by a factor of 1.44, which is a 44% increase.

Generalization

For a circle with an arbitrary original radius ( r ), if the radius is increased by 20%, the new radius is ( 1.2r ). The new area is:

( A_{new} pi (1.2r)^2 1.44 pi r^2 )

The increase in area can be calculated as follows:

( frac{A_{new}}{A_{original}} frac{1.44 pi r^2}{pi r^2} 1.44 )

The area increases by:

( (1.44 - 1) times 100% 44% )

Conclusion

The area of a circle increases by a factor of 1.44 when the radius is increased by 20%. This relationship holds true for any initial radius, as the increase is proportional to the square of the radius change.

Additional Insights

The increase in area can also be seen as a function of the increase in linear dimensions. For example, if the radius is increased from 1 unit to 1.15 units:

( A_{1.15} pi (1.15)^2 pi times 1.3225 1.3225 pi )

This shows a 32.25% increase in area:

( (1.3225 - 1) times 100% 32.25% )

This illustrates how the area scales with the square of the radius.