Regular Polygon Apothem CalculatorFree online math calculator — instant results, no sign-up
Calculate a regular polygon's apothem from its side length.
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About the Regular Polygon Apothem Calculator
The Regular Polygon Apothem Calculator finds the perpendicular distance from a regular polygon’s center to the midpoint of a side using its number of sides and side length. For the default inputs of 6 sides and a side length of 4, it returns an apothem of 3.4641016151 in the same length unit.
- Number of sides
- The unitless count of sides in the regular polygon must be an integer of at least 3 and defaults to 6.
- Side length
- The length of each equal side must be positive, defaults to 4, and may be expressed in any consistent length unit, such as centimeters or inches.
Formula and method
a = s / [2 × tan(π / n)]; with n = 6 and s = 4, a = 4 / [2 × tan(π / 6)] = 2√3 ≈ 3.4641016151.
- aThe apothem, measured in the same length unit as the side length.
- sThe length of one side of the regular polygon.
- nThe number of sides, an integer of at least 3.
- πPi, the circle constant; π/n is an angle measured in radians.
- tanThe tangent function, evaluated using radians in this formula.
Connecting the center to two adjacent vertices forms an isosceles triangle. The apothem bisects this triangle into two right triangles, each with an angle of π/n at the center, an opposite side of s/2, and an adjacent side of a. Therefore, tan(π/n) = (s/2)/a. The calculator evaluates π/n, takes its tangent, and divides the side length by twice that tangent. With the default inputs, the exact result is 2√3, displayed as 3.4641016151.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Number of sides6
- Side length4
Results
- Result3.4641016151
How to use the Regular Polygon Apothem Calculator
- 1Confirm that the polygon is regular: all sides and all interior angles must be equal.
- 2Enter the Number of sides; leave the default value of 6 to calculate for a regular hexagon.
- 3Enter the Side length in your chosen length unit; use the default value of 4 for the worked example.
- 4Run the calculation to evaluate a = s / [2 × tan(π/n)].
- 5Read the result as the center-to-side perpendicular distance: the default inputs give 3.4641016151 in your chosen length unit.
Reading your result
The Result row gives the apothem, not the distance from the center to a vertex. A result of 3.4641016151 means that the perpendicular distance from the center to any side of the regular hexagon is 3.4641016151 length units when each side is 4 units long. The apothem is also the radius of the circle inscribed in the polygon.
Limitations
The formula assumes a regular, convex polygon in a flat plane. It requires an integer side count of at least 3 and a positive side length; unequal sides or unequal angles are outside its scope. The output uses the input length unit without converting units. Decimal results are approximations, and displayed precision does not imply that a measured side length is equally precise.
Common questions
Why does a regular hexagon with side length 4 have an apothem of 3.4641016151?+
For 6 sides, the half-central angle is π/6 radians, or 30 degrees. Substituting into a = s / [2 × tan(π/n)] gives 4 / [2 × tan(π/6)] = 2√3, which rounds to 3.4641016151.
Is the apothem the same as the distance from the center to a vertex?+
No. The apothem ends at the midpoint of a side and is perpendicular to that side. The center-to-vertex distance is the circumradius, which is longer than the apothem.
Can I use this calculator for a polygon that only has equal side lengths?+
Only if its interior angles are also equal. Equal side lengths alone do not generally make a polygon regular, so the side count and side length may not determine its apothem.
How can I use the apothem to find the regular polygon’s area?+
Use A = (n × s × a)/2, where n × s is the perimeter. For the default inputs, A = (6 × 4 × 2√3)/2 = 24√3 square units; using the exact apothem avoids intermediate rounding.
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.