Changes in Radius

Changes in the Radius of a Circle and Its Impact on Area

Relationship Between Radius and Area

• The area (
  $A$
  ) of a circle is directly related to its radius (
  $r$
  ).

• As the radius changes, the area experiences either exponential growth or reduction:

  • Exponential growth: occurs when the radius increases.

  • Exponential reduction: occurs when the radius decreases.

Effects of Multiplying the Radius

• If the radius of a circle is multiplied:

  • By 2: The new area will be 4 times the original area.

  • Example: If the original radius is
    $r$,
    the area changes from:
    $A<em>{ ext{original}} = ext{π} r^2$ to $A</em>{ ext{new}} = ext{π}(2r)^2 = 4 ext{π} r^2$.

  • By 3: The new area will be 9 times the original area.

  • Example: If the original radius is
    $r$,
    the new area becomes:
    $A_{ ext{new}} = ext{π}(3r)^2 = 9 ext{π} r^2$.

Effects of Dividing the Radius

• If the radius of a circle is divided:

  • By 2: The new area will be 1/4 of the original area.

  • Example: If the original radius is
    $r$,
    the new area is:
    $A_{ ext{new}} = ext{π}\bigg( rac{r}{2}\bigg)^2 = rac{1}{4} ext{π} r^2$.

Finding the Area When the Radius Changes

1. Formula for Area:

   • The area of a circle is given by
     $A = ext{π} r^2$.

2. Change Calculation:

   • When the radius changes, calculate the change, multiply it by 2, 4, 6, etc., and then square that number.

     • Example:

     • If the change in radius is represented as a factor,
       $k = 2, 4, 6,…$,

     • New multiplier after squaring is
       $k^2$.

3. Calculate New Area:

   • Multiply or divide the original area by the squared number to find the new area:

     • If radius is multiplied:
       $A<em>{ ext{new}} = A</em>{ ext{original}} imes k^2$.

     • If radius is divided:
       $A<em>{ ext{new}} = A</em>{ ext{original}} imes rac{1}{k^2}$.

Example Calculation

• Initial Radius: 5 cm

  • Calculate Initial Area:

  • $A = ext{π}(5)^2$

  • $A = 78.5 ext{ cm}^2$ (using
    $ext{π} ext{ approximately as } 3.14$).

• New Radius: 20 cm (when the radius is multiplied by 4)

  • Calculate New Area Using Area Formula:

  • $A_{ ext{new}} = ext{π}(20)^2$

  • $A_{ ext{new}} = ext{3.14} imes 400 = 1256 ext{ cm}^2$.

  • Calculate New Area Using Multiplier:

  • Original area multiplied by the change squared:

    • Change Factor:
      $k = 4, ext{ thus } k^2 = 16$.

    • New area:

    • $A_{ ext{new}} = 78.5 ext{ cm}^2 imes 16$

    • $A_{ ext{new}} = 1256 ext{ cm}^2$.

Conclusion

• This illustrates how changes in the radius of a circle can dramatically affect its area, emphasizing the importance of understanding the mathematical principles behind these relationships.