Polygon Inradius Calculator
Enter the number of sides and side length to find the inradius.
Result
How it is calculated
For a regular polygon with n sides of length s (all sides and angles equal), the tool computes the inradius r = s ÷ (2 × tan(π ÷ n)), the circumradius R = s ÷ (2 × sin(π ÷ n)), the perimeter P = n × s and the area A = ½ × P × r. The angle π ÷ n is in radians. The side length can be in any unit, and the radii come out in that unit while the area is in its square. Results show four decimals. The number of sides must be at least 3; a non-whole number is rounded down (6.9 is treated as 6), and the side length must be above zero. Irregular polygons are not supported.
Worked example
Take a regular octagon (n = 8) with side length 10. The angle π ÷ 8 is 22.5°, where tan = 0.41421 and sin = 0.38268. Inradius r = 10 ÷ (2 × 0.41421) = 12.0711. Circumradius R = 10 ÷ (2 × 0.38268) = 13.0656. Perimeter = 8 × 10 = 80, so area = ½ × 80 × 12.0711 = 482.8427.
Common mistakes
- Applying the formula to an irregular polygon. All sides and angles must be equal, otherwise there is no single inradius or centre.
- Entering the distance across the flats or across the corners as the side. A wrench size measures flat to flat, which is twice the inradius, so enter the actual edge length.
- Confusing inradius with circumradius. The first touches the middle of each side, the second reaches the corners, and R is always larger.
- Using degrees for the number of sides, such as 45 for an octagon's angle. The field takes how many sides there are (8), not an angle.
- Entering a fractional side count and expecting interpolation; the tool rounds it down to a whole number.
About this calculator
The inradius (apothem) of a regular polygon is the distance from its center to the midpoint of any side — equivalently, the radius of the largest circle that fits entirely inside the polygon. For a regular polygon with n sides of length s, it's calculated as r = s / (2 × tan(π/n)), which shrinks toward the circumradius as n grows large and the polygon approaches a circle.
The apothem shows up directly in the standard polygon area formula, Area = (1/2) × perimeter × apothem, which is why it matters in fields like machining and CNC programming (fitting a hex bolt head into a socket), carpentry and fabrication (cutting regular polygon panels or gazebo bases), and packaging design (nesting circular components inside hexagonal or octagonal trays).
This calculator takes the number of sides and side length of a regular polygon and returns its inradius, circumradius, and area in one step — useful for engineers, hobbyist woodworkers, and geometry students who need the apothem without re-deriving trigonometric identities each time.
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