Regular Polygon Side CalculatorFree online math calculator — instant results, no sign-up
Calculate a regular polygon's side from its circumradius.
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About the Regular Polygon Side Calculator
The Regular Polygon Side Calculator finds the length of one side of a regular polygon from its number of sides and circumradius. The circumradius is the distance from the polygon’s center to any vertex, not to the midpoint of a side. With the default 6 sides and circumradius of 5, the calculator returns a side length of 5.
- Number of sides
- The number of equal sides in the regular polygon is a unitless whole number of at least 3, with a default of 6.
- Circumradius
- The circumradius is the distance from the polygon’s center to any vertex, entered in your chosen length unit, with a default of 5.
Formula and method
s = 2R × sin(π/n). With the default inputs: s = 2 × 5 × sin(π/6) = 10 × 0.5 = 5.
- sLength of one polygon side, in the same unit as the circumradius.
- RCircumradius: the distance from the center to a vertex.
- nNumber of sides of the regular polygon.
- πPi; π radians equals 180 degrees.
Adjacent vertices subtend a central angle of 2π/n radians. Joining them to the center creates an isosceles triangle; bisecting it gives a right triangle with hypotenuse R, opposite side s/2, and angle π/n. Thus, s = 2R × sin(π/n). The calculation evaluates this sine in radians and multiplies it by twice the circumradius.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Number of sides6
- Circumradius5
Results
- Result5
How to use the Regular Polygon Side Calculator
- 1Enter the Number of sides as a whole number of at least 3; use 6 for the default example.
- 2Enter a positive Circumradius in your chosen length unit; use 5 for the default example.
- 3Calculate the half central angle: π/n = π/6 radians.
- 4Apply the side-length formula: 2 × 5 × sin(π/6) = 5.
- 5Read the Result as the length of each side, using the same unit as the circumradius.
Reading your result
The Result row gives the length of one side, not the perimeter or diameter. For Number of sides = 6 and Circumradius = 5, Result = 5 means every side of the regular hexagon is 5 length units long.
Limitations
The formula assumes a regular, convex polygon in flat Euclidean geometry, with equal sides and equal interior angles. The number of sides must be an integer of at least 3, and the circumradius must be positive for a nondegenerate polygon. The radius must reach a vertex; substituting the apothem, which reaches a side’s midpoint, produces the wrong result. Displayed values may be rounded, and output precision depends on the accuracy of the supplied circumradius.
Common questions
Why does the default regular hexagon have a side length equal to its circumradius?+
For 6 sides, the half central angle is π/6 radians and its sine is 0.5. The formula therefore gives s = 2 × 5 × 0.5 = 5, equal to the circumradius.
Can I enter the apothem instead of the circumradius?+
No. The apothem measures from the center to a side’s midpoint, while this calculator requires the distance to a vertex. If you know the apothem a, convert it first using R = a/cos(π/n).
Does adding more sides increase or decrease the side length?+
With the circumradius held fixed, adding more sides decreases each side’s length because sin(π/n) decreases as n increases.
Should I use radians or degrees when checking the formula?+
Use radians for s = 2R × sin(π/n). If your calculator is in degree mode, use the equivalent formula s = 2R × sin(180°/n).
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.