To find the area of a circle in terms of its circumference, we start with the formulas for circumference and area.
-
The circumference \( c \) of a circle is given by: \[ c = 2\pi r \] where \( r \) is the radius.
-
From this, we can solve for the radius \( r \): \[ r = \frac{c}{2\pi} \]
-
The area \( a \) of a circle is given by: \[ a = \pi r^2 \]
-
Substituting the expression for \( r \) into the area formula: \[ a = \pi \left(\frac{c}{2\pi}\right)^2 \]
-
Simplifying this expression: \[ a = \pi \cdot \frac{c^2}{(2\pi)^2} \] \[ a = \pi \cdot \frac{c^2}{4\pi^2} \] \[ a = \frac{c^2}{4\pi} \]
So, the area \( a \) of the circle in terms of its circumference \( c \) is: \[ a = \frac{c^2}{4\pi} \]
The correct response is: a = \(\frac{c^2}{4\pi}\).
1 answer