How to Find the Length of the Sides of an Octagon Based on Diameter

Stop signs are regular octagons with both types of diameters.
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An octagon can have two types of diameters. Both diameters result from a regular octagon, in which each side is equal in length and each angle between two intersecting sides measures 135 degrees. One type of diameter measures the perpendicular distance between two parallel sides, with half of this diameter equaling the shape's apothem, also called its inradius. The other type measures the distance from opposite angles and separates the octagon into two equal halves, and half of this diameter composes the shape's radius, also known as its circumradius. Both the apothem and circumradius map out circles that either inscribe or circumscribe the octagon — the apothem aids in inscribing a circle inside the octagon, while the circumradius helps plot a circle that surrounds the shape. Each diameter type can produce one of the octagon's identical sides with the aid of trigonometric functions and the mathematical constant pi, which has an approximated value of 3.142.

Apothem or Inradius

    Divide pi by 8 with your calculator. Pi divided by 8 results in approximately 0.393.

    Calculate the tangent value of 0.393 in radians with your calculator. The tangent function is generally denoted by "tan." The tangent of 0.393 in radians approximately equals 0.414 radians.

    Multiply the diameter, which is the perpendicular length between two parallel sides, by 0.414 to calculate a side's length. For example, the diameter measures 5 inches and 5 inches multiplied by 0.414 equals 2.07 inches.

Radius or Circumradius

    Divide pi by 8 with your calculator. Pi divided by 8 results in approximately 0.393.

    Calculate the sine value of 0.393 in radians with your calculator — the sine function is generally denoted as "sin." The sine of 0.393 in radians approximately equals 0.383 radians.

    Multiply the diameter's length, the distance from vertex to opposite vertex, by 0.383 to calculate a side's length. For example, the diameter is 10 inches — 10 inches multiplied by 0.383 results in 3.83 inches.

    Tips

    • Check your math with an online regular polygon calculator. Select eight for the number of sides, and halve each type of diameter before inputting them to obtain the correct lengths for the apothem and circumradius.

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