Hexagon Calculator

What This Calculator Does

You need the area of a hexagonal tile, the perimeter of an octagonal gazebo, or the apothem of a pentagonal bolt head. This regular polygon calculator handles all of those and any other regular polygon from 3 to 12 sides and beyond. Enter the number of sides and the side length, and the tool returns the area, perimeter, apothem, circumradius, and interior and exterior angles. It defaults to a hexagon (6 sides), which is the most searched regular polygon after the square.

A regular polygon has all sides equal and all angles equal. The formulas generalize across every shape in that family, from the equilateral triangle up through the square, pentagon, hexagon, octagon, and dodecagon. For the special case of a triangle, you can cross-check with our Triangle Calculator.

Inputs Required

• Number of sides (n): Any integer from 3 upward. Use the quick-pick buttons for common shapes.
• Side length (s): The length of one side of the polygon

Outputs Provided

• Area: The space enclosed by the polygon
• Perimeter: The total length of all sides
• Apothem: The distance from the center to the midpoint of any side
• Circumradius: The distance from the center to any vertex
• Interior and Exterior Angles: The angles at each vertex and the turn angle

How the Calculation Works

The area formula comes from splitting the polygon into n identical isosceles triangles, each with a base of s and a height equal to the apothem. One triangle has an area of ½ × s × apothem, and multiplying by n gives the total area. Substituting the apothem formula and simplifying produces the single-formula version A = (n × s²) / (4 × tan(π/n)). The apothem is the distance from the center to the midpoint of a side, and the circumradius is the distance from the center to a corner. The two are related through the right triangle formed by half a side, the apothem, and the circumradius.

For a hexagon specifically, the formulas simplify nicely. The apothem of a regular hexagon is s × √3 / 2, the area is (3√3 / 2) × s², and the circumradius equals the side length. That last property, that the radius equals the side, is what makes hexagons tile so efficiently and appear so often in nature, from honeycomb cells to basalt columns. For general area work across shapes, see our Area Calculator.

How to Use the Calculator

1. Pick a shape using the quick-pick buttons, or type the number of sides directly. The minimum is 3.
2. Enter the side length. Use any unit (inches, centimeters, meters) as long as you are consistent.
3. Read the area, perimeter, apothem, circumradius, and angles from the results panel.
4. The formulas with substituted values appear below the result cards so you can verify each step.

Example Calculations

Example 1: A Hexagonal Tile Floor

A homeowner tiles a bathroom with hexagonal tiles that have a side length of 4 inches. The area of one tile is (6 × 16) / (4 × tan(30°)) = 96 / (4 × 0.5774) ≈ 41.57 square inches. The apothem is 4 / (2 × tan(30°)) ≈ 3.46 inches, and the perimeter is 24 inches. To cover 40 square feet (5760 square inches), the homeowner needs about 5760 / 41.57 ≈ 139 tiles, plus 10% waste for cuts, so 153 tiles total.

Example 2: An Octagonal Gazebo

A carpenter builds an octagonal gazebo with 8 sides of 5 feet each. The area is (8 × 25) / (4 × tan(22.5°)) = 200 / (4 × 0.4142) ≈ 120.71 square feet. The apothem is 5 / (2 × tan(22.5°)) ≈ 6.04 feet, which is the distance from the center to the midpoint of each wall. The circumradius is 5 / (2 × sin(22.5°)) ≈ 6.53 feet, which sets the radius for laying out the foundation posts. The interior angle is (8 − 2) × 180 / 8 = 135°, so each corner joint needs a 22.5° miter cut on both boards.

Real World Scenarios

Sizing a Hexagonal Bolt Head

A mechanic needs to know the across-flats dimension of a hex bolt with a 10mm side length. The apothem is 10 / (2 × tan(30°)) ≈ 8.66mm, so the across-flats distance (twice the apothem) is about 17.32mm. That means a 17mm socket is slightly too small and an 18mm socket fits. The across-corners distance (twice the circumradius) is 2 × 10 = 20mm, which matters when the bolt sits in a recessed hole that must clear the corners.

Designing a Pentagon-Shaped Plot

A landscape architect lays out a pentagonal garden plot with 12-foot sides. The area is (5 × 144) / (4 × tan(36°)) = 720 / (4 × 0.7265) ≈ 247.75 square feet. The apothem is 12 / (2 × tan(36°)) ≈ 8.26 feet, which is the radius of the inscribed circle and the distance to set when staking out the center point. The circumradius is 12 / (2 × sin(36°)) ≈ 10.21 feet, which sets the outer stake positions. The interior angle is (5 − 2) × 180 / 5 = 108°, so the edging pieces meet at 108° corners.

Calculating Material for a Dodecagonal Clock Face

A woodworker builds a 12-sided clock face with 4-inch sides. The area is (12 × 16) / (4 × tan(15°)) = 192 / (4 × 0.2679) ≈ 179.14 square inches. The perimeter is 48 inches, which is the length of trim needed for the edge. The circumradius is 4 / (2 × sin(15°)) ≈ 7.73 inches, so the clock face is about 15.46 inches across at the corners. The interior angle is (12 − 2) × 180 / 12 = 150°, meaning each of the 12 segment joints needs a 15° miter cut.

Common Mistakes to Avoid

• Confusing apothem with circumradius: The apothem is the distance from the center to the midpoint of a side, while the circumradius is the distance from the center to a vertex. The circumradius is always larger. For a hexagon, the circumradius equals the side length, but the apothem is s × √3 / 2.
• Using degrees instead of radians in the tangent: The formula uses tan(π/n), which is in radians. If you compute tan(180/n) on a calculator set to degree mode, you get the right answer, but mixing modes produces wrong results. This calculator handles the conversion for you.
• Forgetting that the polygon must be regular: These formulas only work for regular polygons where all sides and angles are equal. For an irregular hexagon, you must split it into triangles and sum their areas individually.
• Entering fewer than 3 sides: A polygon needs at least 3 sides. The calculator rejects inputs below 3. If you need triangle-specific tools like angle solving, use the Triangle Calculator instead.

Limitations of This Calculator

This tool handles regular polygons only, where all sides and all angles are equal. It does not compute the area of irregular polygons, concave polygons, or polygons with varying side lengths. For irregular shapes, decompose the polygon into triangles and rectangles, then sum the parts. The calculator also does not compute the area of a polygon from vertex coordinates; for that, use the shoelace formula directly.

Authoritative Research and Resources

• Wolfram MathWorld: Regular Polygon - A formal reference covering the general formulas for area, apothem, circumradius, and angles of any regular n-gon, with derivations.
• Maths Is Fun: Regular Polygons - A visual guide to regular polygon properties with interactive diagrams showing how the apothem and circumradius relate to the side length.

For related tools, calculate triangle properties with our Triangle Calculator, general shape areas with our Area Calculator, or ellipse area with our Ellipse Calculator.