Question

what is the ratio of the sector area to the area of the entire circle? options: $\frac{1}{5}$, $\frac{1}{2}$, 2, 4. (image shows a circle with center n, radius 4, sector with central angle 72° between points l, n, m.)

Explanation:

Step1: Recall the formula for the ratio of sector area to circle area.

The ratio of the sector area to the area of the entire circle is equal to the ratio of the central angle of the sector to the total angle in a circle (which is \(360^\circ\)). Let the central angle of the sector be \(\theta\) and the total angle in a circle be \(360^\circ\). So the ratio \(r=\frac{\theta}{360^\circ}\).

Step2: Substitute the given central angle.

Here, the central angle of the sector \(\theta = 72^\circ\). Substituting into the formula, we get \(r=\frac{72^\circ}{360^\circ}\).

Step3: Simplify the fraction.

Simplify \(\frac{72}{360}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 72. \(\frac{72\div72}{360\div72}=\frac{1}{5}\).

Answer:

\(\frac{1}{5}\) (corresponding to the option \(\boldsymbol{\frac{1}{5}}\))