Area of a Regular Polygon Calculator

How to use the regular polygon area calculator

What this calculator does

This calculator solves the core geometry of a regular polygon: a flat, convex shape whose sides have equal length and whose interior angles are equal. Give it the number of sides and one known measurement – Side a, Circumradius, Inradius/apothem, or Area – and it derives the remaining dimensions. It is useful for checking classroom work, sizing regular tiles and panels, estimating material coverage, or converting between a drawing's radius and side length. It does not calculate irregular polygons, star polygons, curved boundaries, manufacturing allowances, or three-dimensional volume.

When to use it

Use it when you know one reliable dimension of an equilateral, equiangular polygon and need the rest. Typical cases include finding the area of a hexagonal paver from its side length, obtaining the side length of a regular sign from its circumradius, checking the apothem needed for layout from a centerline, or reverse-solving a regular polygon that must fit a target area. For the underlying definition and broader identities, see Wolfram MathWorld's regular polygon reference.

How to calculate

1. The calculator opens with a complete demonstration: 12 sides, Side a selected, a known value of 5, and inches. The results and example Excel workbook are available immediately.
2. Replace Number of sides with a whole number from 3 through 10,000. Then choose the dimension you know in Known measurement.
3. Enter the positive Known value. The displayed unit beside this field changes automatically: it is a linear unit for a side or radius and a square unit for area.
4. Choose the Length unit. Changing units converts the current value instead of merely relabeling it. Read the live area, radii, perimeter, angles, and detail table.
5. Select Download Excel to create a current-state OOXML workbook with Summary, Inputs, and Geometry sheets. Select Reset to clear the demonstration and calculated state. After Reset, Excel export remains unavailable until a complete valid input set is entered again.

Input guide

Number of sides is required and must be an integer from 3 to 10,000; for example, 12 describes a dodecagon. Increasing this number while holding a side length fixed increases the perimeter and area, while the shape approaches a circle. A common mistake is entering a fractional count or using 2, which cannot define a polygon. Known measurement is required and tells the calculator what the numeric value represents. Choose Side a for an edge, Circumradius for the center-to-vertex distance, Inradius/apothem for the perpendicular center-to-side distance, or Area for covered surface. Choosing the wrong interpretation produces a mathematically valid but physically wrong result.

Known value is required, must be greater than zero, and accepts a decimal point with optional correctly grouped thousands commas, such as 5, 2.75, or 1,250.5. Scientific notation and decimal commas are rejected to avoid silent reinterpretation. For a linear measurement, a larger value scales all lengths proportionally and area by the square of that change. For area, the reverse-solved lengths scale with the square root of the entered area. Length unit is required and can be millimeters, centimeters, meters, inches, feet, or yards. It applies to all linear outputs; the area appears in the corresponding square unit. The selected unit must match the physical measurement you entered. NIST explains why area is expressed in square units such as square meters.

Output guide

Area is the primary result and measures the two-dimensional region inside the polygon in square units. Side a is one edge length. Circumradius runs from the center to a vertex, while Inradius/apothem runs perpendicularly from the center to the midpoint of a side. Perimeter is the total boundary length. Central angle is the angle between adjacent vertex radii, and Interior angle is the angle inside each corner. All are exact mathematical consequences of the regular-polygon assumptions before display rounding; they are not construction tolerances or recommendations. A zero output is not valid here because all accepted known measurements are positive. Very high side counts are valid, but the polygon becomes visually close to a circle and small rounding differences may become more noticeable in displayed text.

The summary pills repeat the current side count, selected known measurement, and unit pair. The Geometry details table lists each output, its formatted value, and the formula route used. The table and workbook read from the same canonical calculation model, so changing an input updates all three presentations together.

Worked example

In the startup example, the polygon has n = 12 sides and a = 5 in. The side-based formula is A = n × a² ÷ [4 × tan(π/n)]Substitution gives A = 12 × 5² ÷ [4 × tan(π/12)] = 279.9038..., displayed as 279.904 in². The perimeter is 12 × 5 = 60 in. The apothem is 5 ÷ [2 × tan(π/12)] = 9.33 in, and the circumradius is 5 ÷ [2 × sin(π/12)] = 9.659 in. OpenStax gives a complementary derivation using one-half the apothem times the perimeter, which produces the same area.