Polygon Area & Perimeter Calculator

Q: How do I find the area and perimeter of a regular polygon?

Enter its side count n and side length s. The calculator uses P=ns, apothem a=s/[2 tan(π/n)] and A=aP/2. All sides and interior angles must be equal.

Q: Can I use this as a regular pentagon or hexagon area calculator?

Yes. Choose regular polygon and enter n=5 for a pentagon or n=6 for a hexagon, then enter the common side length.

Q: Why does a trapezoid need a horizontal offset?

Both bases and height determine area but do not determine perimeter. The signed offset fixes the placement of the upper base, allowing both legs to be derived.

Q: What is the difference between a trapezoid and a trapezium here?

This tool uses trapezoid for a quadrilateral with a pair of parallel bases, commonly called a trapezium in the UK. Equal bases are accepted as the parallelogram limiting case.

Q: Can I enter an irregular polygon or vertex coordinates?

No. Supported shapes are regular polygons, parallelograms, rhombi and trapezoids defined by the displayed dimensions. Arbitrary coordinates, holes and self-intersecting polygons are outside scope.

Q: Does changing units preserve the shape?

Yes. Numeric length entries are converted; side count stays unchanged. Area uses squared units and perimeter uses ordinary length units.

Polygon area and perimeter from defining dimensions

Area measures the region inside a flat shape, in square units. Perimeter measures the length of its boundary, in ordinary length units. This polygon area and perimeter calculator supports convex regular polygons, parallelograms, rhombi and trapezoids. Before calculation, the reference sketch identifies the dimensions. After Calculate, it redraws the chosen shape from your measurements, fits it uniformly into the diagram box and labels the dimensions used in the answer. Editing an input restores the reference sketch until you calculate again.

A regular polygon has equal sides and equal interior angles. Choose n = 5 for a regular pentagon, n = 6 for a regular hexagon, or another whole number from 3 to 1,000. An irregular polygon cannot generally be determined by its side count and one side. For individual circle or triangle measurements, use the existing Circle Calculator or Triangle Calculator.

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Optional geometry sets (paid link) can help draw the base, diagonals and perpendicular height. No purchase is required to use the calculator.

How to use the polygon area and perimeter calculator

Select the shape and one length unit for all measurements.
For a regular polygon, enter the whole-number side count and one side length.
For a parallelogram, enter its base, adjacent side and perpendicular height. Height cannot exceed the adjacent side.
For a rhombus, enter the two full diagonal lengths.
For a trapezoid, enter both parallel bases, their perpendicular separation and the signed horizontal offset of the upper base.
Calculate, then read area, perimeter, derived dimensions and the input-based steps.

Switching units converts numeric length entries so the represented size stays the same; the side count is unchanged. Changing any input clears the old answer. Inputs hidden by another shape are ignored. Reset returns to regular polygon, six sides and centimetres with blank length fields.

Optional graph paper notebooks (paid link) can help keep linear and square units clear during a manual check. Scientific notation such as 1e-3 is accepted for lengths; commas, fractions and expressions are not parsed.

Area and perimeter formulas

Regular polygon: P = ns; a = s/[2 tan(π/n)]; A = aP/2
Parallelogram: A = bh; P = 2(b + s)
Rhombus: A = d1d2/2; s = √[(d1/2)² + (d2/2)²]; P = 4s
Trapezoid: A = (a + b)h/2
Left leg = hypot(h, x); right leg = hypot(h, x + b − a)
Trapezoid P = a + b + left leg + right leg

For a regular polygon, a in its formula is the apothem: the perpendicular distance from the centre to a side. The angle π/n is in radians internally. Do not substitute a diagonal or circumradius for the side input. For the parallelogram, b is the selected base and s is its adjacent sloping side; the height is perpendicular to b.

For the trapezoid only, a and b mean the lower and upper parallel bases, and x is the signed offset. The bottom base runs from (0, 0) to (a, 0). The upper base runs from (x, h) to (x + b, h). Those four positions determine both legs, so the tool can calculate a consistent perimeter rather than guessing missing side lengths.

Worked regular polygon and quadrilateral examples

A regular hexagon with side 4 cm has perimeter 6 × 4 = 24 cm. Its apothem is 4/[2 tan(π/6)] ≈ 3.46410162 cm, giving area ≈ 41.5692194 cm². A regular pentagon or another n-sided shape uses the same formula with its own n.

For a parallelogram with base 8 cm, adjacent side 5 cm and perpendicular height 3 cm, area is 8 × 3 = 24 cm² and perimeter is 2(8 + 5) = 26 cm. Using the sloping side in place of height would give the wrong area.

A rhombus with diagonals 6 cm and 8 cm has side √(3² + 4²) = 5 cm. Its area is 6 × 8/2 = 24 cm² and perimeter is 20 cm.

For a trapezoid with lower base 10 cm, upper base 6 cm, height 3 cm and offset 2 cm, both legs are √13 cm. Area is 24 cm² and perimeter is 16 + 2√13 ≈ 23.2111026 cm. Offset zero makes the left leg perpendicular; a negative offset means the upper base starts to the left.

Understand square units, perimeter and offset

The large result is area; the perimeter card is a separate length. Converting 1 cm to 10 mm multiplies lengths by 10 and areas by 100. A physically unchanged shape therefore has different numeric measurements in different units. Values are displayed to up to nine significant digits; the calculation uses unrounded intermediate values.

The signed trapezoid offset can change the two legs and perimeter while leaving the parallel bases and height unchanged. Its area stays the same. The definition here includes the equal-base limiting case, which is a parallelogram; some school curricula reserve the word trapezoid for exactly one pair of parallel sides. In UK usage the corresponding shape is commonly called a trapezium.

Optional protractors (paid link) can support angle practice, but this tool derives regular-polygon angles from n. For a separate conversion between area units, use the Area Converter.

Supported shapes, precision and limitations

Each required positive length must represent 10⁻⁹ to 10⁹ metres in the chosen unit. The trapezoid offset may be zero or negative, with magnitude no greater than 10⁹ metres. Side counts must be integers from 3 to 1,000. Derived dimensions may exceed the input range. Degenerate zero-area shapes and a parallelogram height greater than its adjacent side are rejected.

The calculator uses standard floating-point arithmetic and trigonometry, so the displayed area of a regular polygon is an approximation, not an exact radical expression. It does not accept vertex lists, arbitrary irregular/self-intersecting polygons, holes, geodesic coordinates or survey bearings. A parallelogram can have mirror-image placements with the same entered lengths; those placements have the same area and perimeter.

Optional geometry workbooks (paid link) can provide practice with these distinctions. For project quantities, the Square Footage Calculator and Paver Calculator have different assumptions. This page adds no waste, cutting plan, joint allowance, price or material recommendation.