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

Central Angles and Inscribed Angles
Central Angles and Inscribed Angles

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 given 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.

Calculating the Central Angle

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

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

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

Measure from a point on the circle to the central angle’s vertex to determine the radius of the circle.

Multiply the radius by pi, a constant that is equal to approximately 3.14.

Multiply the result by 2 to complete the circumference calculation.

Calculating Arc Length

  1. Protractor
  2. Ruler
  3. Calculator

Calculate the circle’s circumference.

Calculate the central angle of your circle, using the protractor, then represent this angle as a fraction. As there are 360 degrees in all circles, make 360 the denominator of the fraction. The angle measurement is the numerator.

Divide the numerator by the denominator to place the number in decimal form.

Multiply the circumference by the decimal to learn the arc length of that section of the circle.

Things You’ll Need


About the Author

Based in Halifax, Nova Scotia, Jordan Whitehouse has been writing on food and drink, small business, and community development since 2004. His work has appeared in a wide range of online and print publications across Canada, including Atlantic Business Magazine, The Grid and Halifax Magazine. Whitehouse studied English literature and psychology at Queen’s University, and book and magazine publishing at Centennial College.

Photo Credits

Kelly Lawrence/Demand Media

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