Polygon Calculator
Polygon calculator for regular and irregular polygons. Compute area, perimeter, apothem, circumradius and interior angles for regular polygons, or use coordinates to find the area of any simple polygon.
Full original guide (expanded)
Polygon Calculator
Use this polygon calculator to compute area, perimeter, interior angles, circumradius and apothem of regular polygons, or switch to the coordinate mode to find the area and perimeter of any simple (non self-intersecting) polygon from its vertex coordinates.
Educational use only – for engineering, CAD or surveying applications, always double-check results, units and tolerances with professional tools.
Regular polygon – area, perimeter, angles
Enter the number of sides and at least one known quantity (side length, perimeter, apothem, circumradius or area). The calculator will fill in the remaining properties of the regular polygon.
Integer, \( n \ge 3 \). Examples: 3 = triangle, 4 = square, 6 = hexagon.
In any length unit (cm, m, in, ft, …).
Distance from centre to any vertex.
Regular polygon results
- Number of sides:
- Side length:
- Perimeter:
- Area:
- Apothem:
- Circumradius:
- Sum of interior angles:
- Each interior angle:
- Central angle:
Polygon by coordinates – area & perimeter
Define a polygon by listing its vertices in order (clockwise or counter-clockwise). The calculator uses the shoelace formula for area and sums edge lengths for perimeter.
Between 3 and 20 points.
Polygon results
- Area:
- Perimeter:
- Orientation:
Orientation tells you whether the vertices were entered in clockwise or counter-clockwise order. The geometric area is reported as a positive value regardless of orientation.
Key formulas for regular polygons
A regular polygon has \( n \) equal sides of length \( s \) and equal interior angles. Important derived quantities include the perimeter \( P \), area \( A \), apothem \( a \) (distance from centre to side) and circumradius \( R \) (distance from centre to vertex).
Regular polygon relationships
\[ P = n s \] \[ A = \frac{1}{2} P a = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)} \] \[ a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}, \quad R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \] \[ \text{Sum of interior angles} = (n - 2)\cdot 180^\circ,\quad \text{each interior angle} = \frac{(n - 2)\cdot 180^\circ}{n} \] \[ \text{central angle} = \frac{360^\circ}{n} \]
These formulas are widely used in geometry, tiling, structural design (for example regular bolt patterns on a flange) and graphics. The calculator implements these relationships directly, using your most convenient input as the starting point.
Area of an arbitrary polygon – shoelace formula
For an irregular polygon defined by ordered vertices \( (x_1, y_1), \dots, (x_n, y_n) \), the shoelace formula gives the area:
The same vertex list can be used to compute the perimeter by summing straight-line distances between consecutive points (including the closing edge from the last point back to the first).
Worked example – regular hexagon
Suppose you have a regular hexagon (\( n = 6 \)) with side length \( s = 10 \,\text{cm} \).
- Perimeter: \( P = n s = 6 \times 10 = 60 \,\text{cm} \).
- Apothem: \( a = \dfrac{s}{2 \tan(\pi/6)} = \dfrac{10}{2 \tan(30^\circ)} \approx 8.66 \,\text{cm} \).
- Area: \( A = \dfrac{1}{2} P a \approx 0.5 \times 60 \times 8.66 \approx 259.8 \,\text{cm}^2 \).
- Each interior angle: \( \dfrac{(6 - 2)\cdot 180^\circ}{6} = 120^\circ \).
Enter \( n = 6 \) and either \( s = 10 \) or \( P = 60 \) in the calculator to reproduce these values.
Practical notes and limitations
- Units: The calculator is unit-agnostic; just be consistent. If you enter side length in metres, the perimeter will be in metres and area in square metres.
- Simple polygons only in coordinate mode: The shoelace formula assumes a simple polygon (no self-intersection). For bow-tie shapes, the signed area corresponds to overlapping regions and may not match the intuitive filled-in area.
- Numerical rounding: Trigonometric functions and square roots are computed in standard JavaScript floating-point arithmetic, so very large \( n \) or extreme dimensions may show small rounding differences compared with symbolic calculations.
- Not a CAD or survey package: For layout of structures, parcels or mechanical parts, always cross-check with specialist CAD or GIS software and relevant standards.
Frequently asked questions
How many inputs do I need to specify for a regular polygon?
You must specify at least the number of sides and one geometric quantity such as side length, perimeter, area, apothem or radius. The calculator then derives the remaining values. If you enter multiple quantities that disagree due to rounding or unit errors, the tool prioritises side length, then perimeter, then radius, apothem and area.
Can I mix different units when entering coordinates?
No. The coordinate mode assumes all x and y values are expressed in the same length unit. If your points are measured in metres and centimetres, convert them to a single consistent unit before entering them.
Why does the orientation say clockwise or counter-clockwise?
The sign of the shoelace sum encodes the orientation of the vertex list. Counter-clockwise ordering typically produces a positive signed area, and clockwise ordering produces a negative one. The calculator reports a positive geometric area but still tells you the underlying orientation, which can be important in graphics and computational geometry.
Is this polygon calculator suitable for exam use?
It is primarily a learning and checking aid. Always follow your exam or course policy on calculator use, and show intermediate steps where required. Using this tool alongside manual derivations can help you verify formulas and catch algebra or arithmetic errors before submitting your work.
Formula (LaTeX) + variables + units
P = n s
A = \frac{1}{2} P a = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)}
a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}, \quad R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)}
\text{Sum of interior angles} = (n - 2)\cdot 180^\circ,\quad \text{each interior angle} = \frac{(n - 2)\cdot 180^\circ}{n}
\text{central angle} = \frac{360^\circ}{n}
A = \frac{1}{2} \left| \sum_{i=1}^{n} x_i y_{i+1} - x_{i+1} y_i \right|
Regular polygon relationships \[ P = n s \] \[ A = \frac{1}{2} P a = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)} \] \[ a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}, \quad R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \] \[ \text{Sum of interior angles} = (n - 2)\cdot 180^\circ,\quad \text{each interior angle} = \frac{(n - 2)\cdot 180^\circ}{n} \] \[ \text{central angle} = \frac{360^\circ}{n} \]
\[ A = \frac{1}{2} \left| \sum_{i=1}^{n} x_i y_{i+1} - x_{i+1} y_i \right| \] with the convention that \( (x_{n+1}, y_{n+1}) = (x_1, y_1) \).
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.