The difference between the circumference and the radius of a circle is 37 cm. The area of the circle is

### Concept & Logic Translate the given relationship into an equation to solve for the radius, and then apply the area formula. $$2\pi r - r = 37$$ ### Step-by-Step Solution * Given: $2\pi r - r = 37$. * Factor out $r$: $r(2\pi - 1) = 37$. * Substitute $\pi \approx \frac{22}{7}$: $r \left(2 \times \frac{22}{7} - 1\right) = 37$. $r \left(\frac{44}{7} - 1\right) = 37$. $r \left(\frac{44 - 7}{7}\right) = 37$. $r \left(\frac{37}{7}\right) = 37$. * Solve for $r$: $r = 7$ cm. * Calculate the area of the circle: $A = \pi r^2 = \frac{22}{7} \times 7 \times 7 = 22 \times 7 = 154$ cm². ### Exam Strategy & Shortcut The expression $(2\pi - 1)$ always evaluates to $\frac{37}{7}$ when using $\pi = \frac{22}{7}$. If you memorize that $(2\pi - 1)r = \frac{37}{7}r$, seeing "$37$" as the difference immediately tells you $r = 7$. An $r=7$ circle always has an area of 154. ### Common Pitfall Misinterpreting "difference between the circumference and the radius" as $r - 2\pi r$, which leads to a negative value. Circumference is always larger than the radius. ### Final Answer Therefore, the correct answer is **154 cm²**.