How to Calculate the Arc Length, Central Angle, and Circumference of a Circle

How to Calculate the Arc Length, Central Angle, and Circumference of a Circle thumbnail
Use a protractor to determine a circle's central angle.

Calculating a circle's arc length, central angle, and circumference are not just tasks, but essential skills for geometry, trigonometry and beyond. The arc length is the measure of a section of a circle's circumference; a central angle has a vertex at the center of the circle and the sides that pass through two points on the circle; and circumference is the distance around the circle. The vertex is the center of the circle. Calculating each of these is easy if you have the right tools and you're using the proper formulas.

Things You'll Need

  • Protractor
  • Ruler
  • Calculator
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  1. Calculating the Central Angle

    • 1

      Place the origin of the protractor on the angle's vertex.

    • 2

      Place the base line of the protractor on one of the angle's sides.

    • 3

      Record the number on the protractor where the second side of the angle passes through the edge of the protractor. If the angle is larger than 90 degrees, record the top number; if the angle is smaller than 90 degrees, record the lower number. This is the measurement of your central angle.

    Calculating Circumference

    • 4

      Determine the radius of the circle by drawing a line from one point on the circle to the central angle's vertex, then measuring the length of this line. The calculation for determining a circle's circumference is two times pi times the radius.

    • 5

      Take note that pi, an irrational number, is equal to approximately 3.14. This is the number to use in your equation, unless your calculator has a pi button.

    • 6

      Multiply 2 times 3.14 times the radius to calculate the circumference.

    Calculating Arc Length

    • 7

      Calculate the central angle of your circle, then represent this angle as a fraction. As there are 360 degrees in all circles, allow 360 to be the denominator; add the angle measurement as the numerator.

    • 8

      Represent the fraction as a decimal by dividing the numerator by the denominator.

    • 9

      Calculate the circle's circumference. Then, multiply the circumference by the decimal calculated in Step 2.

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