A circle has circumference 110 cm. Find its area, taking pi to be 22/7.

Step-by-step solution

1. 1

   Recover the radius first. Area needs $r$, and the only thing you are given is the circumference, so start from

   $C=2\pi r=110.$

2. 2

   Solve for r with pi = 22/7. Dividing by a fraction means multiplying by its reciprocal:

   $r=\frac{110}{2\pi}=\frac{110}{2\cdot\frac{22}{7}}=\frac{110}{\frac{44}{7}}=110\cdot\frac{7}{44}=\frac{770}{44}=17.5\ \text{cm}.$

   The value $\tfrac{22}{7}$ is chosen precisely so this comes out exact — $44$ divides $110$ into $2.5$, and $2.5\times 7=17.5$.

3. 3

   Square the radius.

   $r^{2}=17.5^{2}=306.25\ \text{cm}^2.$

4. 4

   Apply the area formula.

   $A=\pi r^{2}=\frac{22}{7}\times 306.25=\frac{22\times 306.25}{7}=\frac{6737.5}{7}=962.5\ \text{cm}^2.$

5. 5

   Check by a shortcut that avoids squaring. Since $C=2\pi r$, the area can be written $A=\tfrac{Cr}{2}$:

   $A=\frac{110\times 17.5}{2}=\frac{1925}{2}=962.5\ \text{cm}^2,$

   the same value with far less arithmetic. This identity — area equals half the circumference times the radius — is worth remembering.