Measure of an arc of a Circle

If the measure of minor arc of a circle is 125 degrees then find the measure of major arc of the circle.

An arc of a circle is a portion of the boundary(circumference) of the circle. If the arc is smaller than the rest of the boundary of the circle, then the arc is called Minor arc. And if it is bigger than the rest of the boundary, the arc is called Major arc. 

Measure of an arc, or simply arc measure, is the angle subtended by that arc at the center of the circle. In other words, arc measure is the measure of its central angle. The boundary of a semicircle subtends angle 180 degrees at the center. The boundary of an entire circle subtends angle 360 degrees at the center. 

In the given example, minor arc of the circle subtends 125 degrees at the center. I will use letters A and P for the end points of this minor arc, and letter O to represent the center of the circle. Connecting the end points of the arc with the center O, the resulting angle ⦣AOP is the angle subtended by the minor arc at the center. 

The rest of the boundary of this circle forms the major arc of the circle. And it also subtends some angle at the center which is x in the above figure. We want to find x. 

The major and minor arc together form the entire boundary of the circle, and hence together they will subtend an angle of 360 degrees at the center. 

x + 125° = 360°

Hence, x = 360° - 125° = 235°

Therefore, the angle subtended by the major arc of the circle at the center is 235 degrees.