Polygon Area Formulas for Regular and Irregular Shapes

Use perimeter and apothem for a regular polygon

A regular polygon can be split into congruent triangles meeting at its centre. Each triangle has a side of the polygon as its base and the apothem as its perpendicular height. Adding their areas gives A = ap/2, where a is the apothem and p is the perimeter. Do not replace the apothem with the circumradius, which reaches a vertex rather than the midpoint of a side. These two distances differ except in a limiting circular approximation.

Work a regular hexagon from its side length

For a regular hexagon with side 4 cm, perimeter is 24 cm. Its apothem is 2√3 cm, approximately 3.4641 cm. Area is (24 × 2√3)/2 = 24√3 cm², approximately 41.5692 cm². Enter six sides and side length four in the linked calculator. Splitting the hexagon into six equilateral triangles gives the same result, providing an independent geometric check on the regular-polygon output.

Derive the general side-length formula

For n sides of length s, the apothem is s/[2 tan(π/n)]. Combining that with p = ns gives A = ns²/[4 tan(π/n)]. The angle π/n is in radians; use 180/n degrees in a degree-mode calculator. This expression assumes a regular polygon. As a simple check, n = 4 gives tan(π/4) = 1 and A = s², the square formula. Doubling s multiplies the calculated area by four because the only length term is squared.

Decompose an irregular polygon instead

For an L-shaped region formed by a 10-by-8 rectangle with a 4-by-3 corner removed, area is 80 − 12 = 68 square units. This uses the actual boundary dimensions without assuming equal sides or angles. Alternatively, split the region into nonoverlapping rectangles and add their areas. Include every part once and subtract only regions outside the intended boundary. An irregular polygon’s perimeter alone is insufficient, and using its average side in the regular formula creates a different hypothetical shape rather than solving the original one.

Check whether the polygon is regular

A polygon has a closed boundary made from straight segments. A regular polygon has equal side lengths and equal interior angles. Equal sides alone do not guarantee regularity: a rhombus can have equal sides but unequal angles. Likewise, a rectangle can have equal angles without equal sides. The regular-polygon calculator takes one side length and a side count, so its area and angle results depend on the regularity assumption. Do not feed an average side length into that form and treat the result as the measured area of an irregular plot.

Keep the measurement types distinct

Side lengths and perimeter use ordinary length units; area uses square units; angles use degrees or radians. If every length in a similar figure is multiplied by k, perimeter multiplies by k and area by k², while angles remain unchanged. This gives a direct check on a calculation with a converted side length. Converting centimetres to metres before calculating should describe the same figure as converting the final square centimetres to square metres, provided the area conversion factor is squared rather than copied from the length conversion.

List enough information to determine the answer

A side count identifies a broad class of shapes, but it does not usually determine their size or area. A perimeter likewise allows many different irregular shapes. Label which conditions are given and which are assumptions, and sketch the boundary before substituting numbers. For coordinate data, preserve the order around the boundary; sorting vertices simply by x-coordinate can create crossing segments. A photograph or drawing is not automatically to scale. Calculator output describes the ideal geometry entered into its fields, so retain the underlying measurements and assumptions when reporting a result.

Related guides

Area from Coordinates: Ordered Vertices and Shoelace

Calculate polygon area from ordered coordinates, work a shoelace example, and check vertex order, orientation and self-intersecting boundaries.

Convex Polygon Formulas: Angles, Diagonals and Area

Apply angle-sum and diagonal formulas to convex polygons, distinguish regular polygons, and explain why side count alone cannot give area.

Identify a Polygon from Sides, Angles or Diagonals

Identify polygon side counts from angle sums, regular exterior angles or diagonal counts, and check that the result is a valid whole number.