Enter the number of sides and side length to compute the apothem and polygon properties
Number of Sides (n)
Side Length (s)
Result
Apothem (a)
-
Circumradius (R)
-
Perimeter
-
Area
-
Detailed Derivation
Apothem & Polygon Formulas
Apothem a = s / (2 x tan(pi/n))
Circumradius R = s / (2 x sin(pi/n))
Perimeter P = n x s
Area A = (P x a) / 2 = (n x s x a) / 2
Central Angle = 360 / n
The apothem is the radius of the inscribed circle of a regular polygon. It is also the perpendicular distance from the center to any side. Together with the circumradius, it fully describes the polygon geometry.
⚠The number of sides must be at least 3. For very large n (n > 100), the polygon approaches a circle and the apothem approximates the circumradius.
What Is an Apothem?
The apothem of a regular polygon is the distance from its center to the midpoint of any side. It acts as the inradius of the polygon and is perpendicular to each side. The apothem is essential for computing the area of regular polygons.
Inscribed Circle
The apothem is the radius of the incircle - the largest circle that fits entirely inside the polygon, touching each side at its midpoint.
Trigonometry
Using the central angle theta = 360/n, and half-angle theta/2: tan(theta/2) = (s/2)/a. Solve for a.
Apothem vs Side
For fixed n, the apothem increases proportionally with side length. For fixed side length, the apothem increases as n increases.
Special Cases
Equilateral triangle (n=3): a = s/(2 x sqrt(3)). Square (n=4): a = s/2. Hexagon (n=6): a = s x sqrt(3)/2.
Teaching Example: Regular hexagon (n=6) with side s=5. Central angle = 360/6 = 60 deg. Half-angle = 30 deg. Tan(30) = 0.577. Apothem = (5/2)/0.577 = 4.330. Circumradius = (5/2)/sin(30) = (2.5)/0.5 = 5. Area = (6x5x4.330)/2 = 64.95.
The apothem is the distance from the center to the midpoint of any side. It is also the radius of the inscribed circle. Formula: a = s/(2 x tan(pi/n)).
What is the apothem of a square?▼
For a square with side length s, the apothem a = s/2. For example, a square with side 10 has apothem = 5. The circumradius R = s/sqrt(2) = 7.07.
Can the apothem equal the circumradius?▼
Never for a polygon with finite sides. As n approaches infinity (circle), the apothem approaches the circumradius. For any finite n, the circumradius is strictly larger.
How is the apothem used in area formulas?▼
Area = (Perimeter x Apothem)/2. This works because the polygon is made of n triangles, each with base = side and height = apothem.
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