Question

the measure of central angle mnl is π radians, and the measure of the entire circle is 2π radians. the ratio of the measure of the central angle to the entire circle measure is dropdown. the area of the entire circle is dropdownπ units². the area of the sector is dropdownπ units². (image shows a circle with center n, radius 6 (nl = 6), sector mnl shaded, points m, l on the circle, n as center.)

Explanation:

First Blank (Ratio of Central Angle to Circle Measure)

Step1: Identify the measures

The central angle \( \angle MNL \) is \( \pi \) radians, and the circle's total measure is \( 2\pi \) radians.

Step2: Calculate the ratio

The ratio is \( \frac{\pi}{2\pi} \). Simplifying, the \( \pi \) terms cancel, giving \( \frac{1}{2} \).

Step1: Recall the area formula for a circle

The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius.

Step2: Determine the radius

From the diagram, the radius \( r = 6 \) units.

Step3: Calculate the area

Substitute \( r = 6 \) into the formula: \( A = \pi (6)^2 = 36\pi \).

Step1: Recall the sector area formula

The area of a sector is the ratio of the central angle to the full circle angle times the area of the circle. The ratio we found earlier is \( \frac{1}{2} \), and the circle's area is \( 36\pi \).

Step2: Calculate the sector area

Multiply the ratio by the circle's area: \( \frac{1}{2} \times 36\pi = 18\pi \).

Answer:

\( \frac{1}{2} \)

Second Blank (Area of Entire Circle)