Question

If the radius of a circle is reduced by 50 percent, by what percent is its area reduced?

Answer 1

Area goes with the square of radius, so if we multiply the radius by `.5` we multiply the area by `.5^2=.25,` a reduction of 75%.

Answer 2

The area will be reduced by `25%`.

Explanation:

Example 1

A circle has a radius of `"4 cm"`. If the radius is reduced by `50%`, what is the new area? By what percent is its area reduced?

Radius `"4.0 cm"`

`A=pir^2`

`A=pi*("4.0 cm")^2="50.3 cm"`

Radius `"2.0 cm"`

`A=pi*("2.0 cm")^2="12.6 cm"^2"`

Percent reduction of area

`(12.6color(red)cancel(color(black)("cm"^2)))/(50.3color(red)cancel(color(black)("cm"^2)))xx100="25.0%`

Example 2

A circle has radius of `"24 cm"`. If the radius is reduced by `50%`, what is the new area? By what percent is its area reduced?

Radius `"24.0 cm"`

`A=pi*(24.0"cm")^2="1809.6 cm"^2"`

Radius `"12.0 cm"`

`A=pi*("12.0 cm")^2="452.4 cm"^3"`

Percent reduction of area

`(452.4color(red)cancel(color(black)("cm"^2)))/(1809.6color(red)cancel(color(black)("cm"^2)))xx100="25.0%"`

I used the `pi` key on my calculator.