Regular Polygon Area Calculator

A regular polygon’s area — hexagon, pentagon, octagon, or any number of sides — can be found from whichever measurement you already know: side length, apothem (inradius), circumradius, perimeter, or even the area itself worked backward. This calculator handles any number of sides (3 or more) from one shared engine, so the same tool covers a triangle, a pentagon, a hexagon, a dodecagon, or anything in between.

Enter the number of sides plus any one known measurement, and the calculator returns the area together with the perimeter, apothem, circumradius, and both the interior and exterior angles — every mode reports all of these at once, not just the single value you started from. This calculator only covers regular polygons, where every side and every angle is equal; an irregular polygon needs its area found a different way, typically from its corner coordinates.

Regular Polygon Area Formula

Area = (n × Side²) ÷ (4 × tan(π/n))

Where:
n = the number of sides
Side = the length of one side (all sides are equal in a regular polygon)
tan(π/n) = the tangent of π/n radians (180°/n)

Worked Example: A Regular Hexagon

A hexagon (n = 6) with 4-inch sides: Area = (6 × 4²) ÷ (4 × tan(30°)) = (6 × 16) ÷ (4 × 0.5774) ≈ 96 ÷ 2.3094 ≈ 41.57 square inches. Perimeter = 6 × 4 = 24 inches.

By Apothem, Circumradius, or Perimeter

Since side length, apothem, circumradius, and perimeter are all related to each other once the number of sides is fixed, this calculator lets you enter any one of them (along with n) and solves for the rest.

By Apothem (Inradius)

Side = 2 × Apothem × tan(π/n)

The apothem is the distance from the center of the polygon to the midpoint of a side — the radius of the largest circle that fits inside the polygon.

Worked Example

A regular pentagon (n = 5) with a 3-ft apothem: Side = 2 × 3 × tan(36°) ≈ 4.36 ft. Area ≈ 32.69 square feet, Perimeter ≈ 21.80 feet, Circumradius ≈ 3.71 feet.

By Circumradius

Side = 2 × Circumradius × sin(π/n)

The circumradius is the distance from the center to any vertex — the radius of the circle that passes through all corners of the polygon.

Worked Example

A regular octagon (n = 8) with a 5-ft circumradius: Side = 2 × 5 × sin(22.5°) ≈ 3.83 ft. Area ≈ 70.71 square feet, Perimeter ≈ 30.61 feet, Apothem ≈ 4.62 feet.

By Perimeter

Side = Perimeter ÷ n

Worked Example

A regular hexagon (n = 6) with a 30-ft perimeter: Side = 30 ÷ 6 = 5 ft. Area ≈ 64.95 square feet.

Finding the Side Length from Area (Reverse)

Side = √(4 × Area × tan(π/n) ÷ n)

This works backward from a known area to find the side length (and from there, the perimeter, apothem, and circumradius) — useful when you know how much space a regular-polygon-shaped patio, garden bed, or floor tile needs to cover and want to know what size to cut it.

Worked Example

A regular hexagon needs to cover 41.57 square feet: Side = √(4 × 41.57 × tan(30°) ÷ 6) = 4 ft — the same hexagon as the worked example above, arrived at in reverse.

Interior and Exterior Angles of a Regular Polygon

Interior Angle = (n − 2) × 180 ÷ n
Exterior Angle = 360 ÷ n

Every mode of this calculator reports both angles alongside the area, perimeter, apothem, and circumradius. A hexagon’s interior angle is 120° and exterior angle is 60°; a regular octagon’s interior angle is 135° and exterior angle is 45° — the two angles at any vertex always add up to 180°, since they sit on a straight line.

Hexagon, Pentagon, Octagon, and Other Common Polygons

Because the number of sides is just one of the inputs, this same calculator handles any named regular polygon — simply set n to the shape you need.

Hexagon Area Calculator (n = 6)

Enter 6 for the number of sides, then any known measurement. Hexagons are especially common in tiling, honeycomb patterns, and nut/bolt-head cross-sections, since regular hexagons tile a flat plane with no gaps.

Pentagon Area Calculator (n = 5)

Enter 5 for the number of sides. Regular pentagons come up in architecture, signage, and geometry problems, though unlike triangles, squares, and hexagons, they cannot tile a flat plane on their own.

Octagon Area Calculator (n = 8)

Enter 8 for the number of sides. Regular octagons are common in stop-sign shapes, gazebo and pavilion floor plans, and decorative tile layouts.

Frequently Asked Questions

How do you find the area of a hexagon?

Set the number of sides to 6, enter the side length (or apothem, circumradius, perimeter, or area), and the calculator returns Area = (6 × Side²) ÷ (4 × tan(30°)). A hexagon with 4-inch sides has an area of about 41.57 square inches.

What’s the difference between apothem and circumradius?

The apothem is the distance from the center to the midpoint of a side (the inradius); the circumradius is the distance from the center to a vertex (corner). The circumradius is always longer than the apothem for the same polygon, since a corner sits farther from the center than the midpoint of an edge.

How do you find a regular polygon’s area from its perimeter?

Divide the perimeter by the number of sides to get the side length, then use the area formula: Area = (n × Side²) ÷ (4 × tan(π/n)). A hexagon with a 30-ft perimeter has 5-ft sides and an area of about 64.95 square feet.

Does this calculator work for irregular polygons?

No — this calculator assumes every side and every angle is equal, which is what makes a polygon “regular.” An irregular polygon, where the sides or angles differ, needs its area calculated a different way, typically from the coordinates of its corners.