HesapHanem

Polygon Inradius Calculator

Enter the number of sides and side length to find the inradius.

Result

12.0711
13.0656
482.8427

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.

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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 approaches 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).

With just the side count and side length of a regular polygon it gives the inradius (apothem), the circumradius and the area together, handy for engineers, hobby woodworkers and geometry students who would rather not rederive the tangent formula each time.

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Comments (1)

  • Editorial team

    Two numbers from this page answer two different shop questions. The circumradius sets the smallest round bar you can turn a regular polygon from, and the inradius sets the largest disc that fits inside it. For a hexagon with 10 mm edges that means at least 20 mm bar across the corners and a disc up to 17.32 mm across. Which workshop job do you use polygons for, cutting panels, machining or packaging?

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