An angle turns through \frac{120}{360} of a circle, as shown below.

An angle turns through \frac{120}{360} of a circle, as shown below. What is the measure of the angle?

The Correct Answer and Explanation is:

To find the measure of the angle that turns through 120360\frac{120}{360}360120​ of a circle, we use the fact that a full circle measures 360 degrees.

Step-by-step Calculation:

Angle=120360×360∘=120∘\text{Angle} = \frac{120}{360} \times 360^\circ = 120^\circAngle=360120​×360∘=120∘

Final Answer: 120 degrees

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Explanation

In geometry, a circle is composed of 360 degrees. When we are told that an angle turns through a fraction of a circle, we multiply that fraction by 360 to determine the actual angle in degrees. In this problem, the angle turns through 120360\frac{120}{360}360120​ of a circle.

To solve this, we multiply:120360×360=120 degrees\frac{120}{360} \times 360 = 120 \text{ degrees}360120​×360=120 degrees

This tells us that the angle formed is 120 degrees.

Visually, this is confirmed in the image. The shaded portion of the circle indicates the angle, and the arrowed lines represent the sides of the angle starting from a common vertex at the center of the circle. Since the shaded region covers about one-third of the circle (as 120 is one-third of 360), the visual representation supports the calculation.

Understanding angles as portions of a circle is a key concept in both geometry and real-world applications like clocks, rotations, and navigation. For instance, a turn of 90 degrees corresponds to one-quarter of a full turn, and 180 degrees is a half-turn. Similarly, 120 degrees means the angle sweeps one-third of the circle.

This also ties into the concept of central angles, where the angle’s vertex is at the center of the circle and its sides are radii extending to the circumference. In such cases, the measure of the angle is directly proportional to the arc it intercepts.

Therefore, by multiplying the fraction of the circle by 360, we accurately calculate the central angle’s degree measure. Here, 120360×360=120∘\frac{120}{360} \times 360 = 120^\circ360120​×360=120∘, which is the correct and final answer.