If a sector of a circle has a central angle of $72°$, what is the ratio of the area of the sector to the area of the entire circle?

• A. $\frac{1}{2}$
• B. $\frac{1}{5}$
• C. $2$
• D. $4$

Answer: (B)

Verified step by step guidance

1. Recall that the area of a sector of a circle is proportional to its central angle compared to the full angle of the circle, which is 360 degrees.
2. Write the formula for the ratio of the area of the sector to the area of the entire circle as the ratio of their angles: \(\frac{\text{Area of sector}}{\text{Area of circle}} = \frac{\text{Central angle}}{360^\circ}\).
3. Substitute the given central angle of 72 degrees into the formula: \(\frac{\text{Area of sector}}{\text{Area of circle}} = \frac{72^\circ}{360^\circ}\).
4. Simplify the fraction \(\frac{72}{360}\) by dividing numerator and denominator by their greatest common divisor.
5. Express the simplified fraction as the final ratio of the sector's area to the entire circle's area.