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 are essential skills for geometry, trigonometry, and beyond. The arc length is how long a certain section of a circle's circumference is; a central angle is an angle that has a vertex at the center of the circle and sides that pass through two points on the circle; and circumference is the distance around a closed curve. 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 one of the angle's sides.

    • 3

      Record the number on the protractor where the second side of the angle passes through the edge 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 2 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. This gives you the arc length.

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