Enter the number of sides and side length to compute area, perimeter, and apothem
Number of Sides (n)
Side Length (s)
Result
Perimeter
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Area
-
Apothem
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Circumradius
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Detailed Derivation
Regular Polygon Formulas
Perimeter P = n x s
Apothem a = s / (2 x tan(pi/n))
Area A = (n x s x a) / 2 = (n x s^2) / (4 x tan(pi/n))
Circumradius R = s / (2 x sin(pi/n))
A regular polygon has all sides equal and all interior angles equal. These formulas work for any regular polygon from triangle to n-gon.
⚠The number of sides must be at least 3. Results use Javascript Math.tan and Math.sin with high precision.
What Is a Regular Polygon?
A regular polygon is a polygon where all sides are equal and all interior angles are equal. Common examples include equilateral triangles, squares, regular pentagons, hexagons, and octagons.
Perimeter
The total distance around the polygon. Simply multiply the number of sides by the side length: P = n x s.
Area
The space enclosed by the polygon. Area = (perimeter x apothem) / 2. Alternatively use the trigonometric formula.
Apothem
The distance from center to side midpoint. Functions as the inradius. Calculated using tangent of the central angle.
Circle Approximation
As n increases, the polygon approaches a circle. At n=100, a regular polygon is virtually indistinguishable from a circle.
Teaching Example: Regular hexagon (n=6) with side s=5. Perimeter = 6x5 = 30. Apothem = 5/(2xtan(pi/6)) = 4.330. Area = (6x5x4.330)/2 = 64.952. Circumradius = 5/(2xsin(pi/6)) = 5.
Use A = (n x s^2) / (4 x tan(pi/n)). For n=6, s=5: A = (6x25)/(4x0.577) = 64.95. Alternatively A = (P x a)/2.
What is the apothem of a regular polygon?▼
The apothem is the distance from center to side midpoint. Formula: a = s/(2 x tan(pi/n)). For a square with side 4: a = 4/2 = 2.
What is the minimum polygon with equal sides?▼
An equilateral triangle (n=3) is the simplest regular polygon. A square (n=4) is next. Regular polygons exist for any n=3 or more.
How does a regular polygon approach a circle?▼
As n increases, the apothem approaches the circumradius, the polygon becomes smoother, and its shape converges to a circle. For n=100, it is nearly indistinguishable from a circle.
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