If the radius of a circle is doubled, how does the area change?

When the radius of a circle is doubled, the area changes in a specific way due to the formula used to calculate the area of a circle. The area ( A ) of a circle is given by the formula:

[

A = \pi r^2

]

where ( r ) is the radius of the circle.

If the radius is doubled, the new radius becomes ( 2r ). Plugging this new radius into the area formula gives:

[

A_{new} = \pi (2r)^2

]

Calculating this leads to:

[

A_{new} = \pi (4r^2) = 4\pi r^2

]

This shows that the new area is four times the original area. Hence, when the radius of a circle is doubled, the area indeed quadruples.

This is conceptually important because it highlights how changes to the linear dimensions of geometric shapes can affect their properties in non-linear ways, particularly in relation to areas. Understanding this relationship is crucial for solving many geometry problems involving circles and other geometric figures.