A circle with a circumference of 111.6 cm is centered at the vertex of an angle. The rays of the angle subtend an arc that is 13.95 cm long. Complete the following statements by entering numbers in the answer boxes. 1/360th of the circumference of the circle is cm long, so the measure of the angle in degrees is . The arc subtended by the angle's rays is % of the circumference of the circle. Since an angle that makes a full rotation measures 360 degrees, the measure of the angle in degrees must be

The [tex]\frac{1}{360}[/tex]th of the circumference is 0.31 centimeters. The arc subtended by the angle's rays is 12.5 % of the circumference of the circle.

How to understand arc and circumferences

In this question we must understand and apply the relationships between arc length and circumference. By geometry we know that central angle of a circle equals 360° and the circle arc is directly proportional to central angle.

The [tex]\frac{1}{360}[/tex]th of the circumference is obtained by multiplying the length of the circumference by [tex]\frac{1}{360}[/tex], then the length of the [tex]\frac{1}{360}[/tex]th of the circumference is:

[tex]s = \frac{1}{360}\times 111.6\,cm[/tex]

[tex]s = 0.31\,cm[/tex]

The [tex]\frac{1}{360}[/tex]th of the circumference is 0.31 centimeters. [tex]\blacksquare[/tex]

The arc subtended by the angle's rays is found by dividing the subtended arc by the arc length:

[tex]n = \frac{13.95\,cm}{111.6\,cm} \times 100\,\%[/tex]

[tex]n = 12.5\,\%[/tex]

The arc subtended by the angle's rays is 12.5 % of the circumference of the circle. [tex]\blacksquare[/tex]

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