Polygon Area Calculator
Calculate the area of any polygon — 3 to 8+ sides, irregular lots, uneven sides, or coordinates. Convert results to acres, hectares, square feet & more.
What is a Polygon Area Calculator?
A polygon is a closed two-dimensional planar geometric figure bounded by three or more straight line segments. In geometry and mathematics, the line segments that form the outer boundary are called sides (or edges), and the intersection points where two adjacent sides meet are called vertices (corners).
Polygons are classified by their number of sides: triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), heptagons (7 sides), octagons (8 sides), nonagons (9 sides), decagons (10 sides), hendecagons (11 sides), and dodecagons (12 sides). Polygons are grouped into two primary structural categories:
In land measurement, real estate parcels and agricultural fields rarely conform to perfect squares or equilateral triangles. Most properties feature irregular boundaries with uneven side lengths, obtuse angles, and curved segments approximated by multi-vertex polygons. The polygon area calculator handles both regular geometric shapes and irregular multi-sided parcels across all common measurement units.
Interactive Polygon Anatomy & Geometry Explorer
Select an anatomy feature and adjust the slider to inspect polygon elements in real time:
How to Use the Polygon Area Calculator
To use the polygon area calculator, follow these five straightforward steps to compute total surface area, perimeter, and unit conversions:
Step 1: Choose Calculation Mode (Side Lengths vs Vertex Coordinates)
Select between Side Lengths Mode and Coordinates Mode depending on your available survey data. Use Side Lengths Mode when you have measured edge lengths of regular polygons or triangles. Use Coordinates Mode when you have Cartesian (X, Y) coordinates, GPS waypoints, or plat map boundary points for irregular lots.
Step 2: Select Number of Sides (3 to 12+ sides) or Add Coordinate Points
Choose the polygon side count from the dropdown menu (3-sided triangle, 4-sided quadrilateral, 5-sided pentagon, 6-sided hexagon, or 8-sided octagon). In Coordinates Mode, use the + Add Point button to define additional boundary vertices for parcels with 9, 12, or more corners.
Step 3: Enter Dimensions with Custom Units (ft, m, yd, in, cm)
Enter the numerical length or coordinate values into each input field. Select your preferred measurement unit from the dropdown selector beside each field. The calculator supports feet (ft), yards (yd), meters (m), centimeters (cm), inches (in), miles (mi), and kilometers (km) with automatic conversion.
Step 4: Toggle Regular Polygon or Input Individual Irregular Side Lengths
Check the Regular polygon checkbox if all sides and angles are equal, which allows you to enter a single side length for the entire shape. Uncheck the box to input individual measurements for each side of an irregular polygon.
Step 5: View Results, Acreage Breakdown, Perimeter, & Interactive SVG Diagram
View your computed polygon area instantly in acres (ac), hectares (ha), square feet (sq ft), square meters (m²), square yards (sq yd), square miles (sq mi), and square kilometers (sq km). Examine the live interactive SVG diagram to visually confirm the shape geometry and vertex sequence.
What is the Formula for Polygon Area? (Calculation Methods)
There are four primary mathematical methods used to calculate the area of a polygon based on the shape classification and available geometric data:
Regular Polygon Area Formula (Side Length & Number of Sides)
For any regular polygon with n equal sides of length s, the area is calculated using the standard trigonometric formula:
Example: A regular pentagon (n = 5) with side length s = 40 ft:
Area = (5 × 40²) ÷ [4 × tan(π ÷ 5)] = (5 × 1,600) ÷ [4 × tan(36°)] = 8,000 ÷ (4 × 0.726543) = 8,000 ÷ 2.90617 = 2,752.76 sq ft (0.06319 acres or 255.74 m²)
Regular Polygon Area Formula Using Apothem & Perimeter
When the apothem (a) — the perpendicular distance from the center of the regular polygon to the midpoint of any side — and the perimeter (P) are known, the area formula is:
Example: A regular hexagon (n = 6) with side s = 30 ft and apothem a = 25.98 ft:
Perimeter P = 6 × 30 ft = 180 ft
Area = ½ × 25.98 ft × 180 ft = 2,338.20 sq ft (0.05368 acres or 217.23 m²)
Shoelace Formula for Coordinates (Gauss's Area / Surveyor's Formula)
The Shoelace formula (also called Gauss's area formula or the Surveyor's formula) calculates the exact area of any 2D polygon from the Cartesian coordinates of its ordered vertices:
Example: A 4-sided irregular parcel with vertices P1(0, 0), P2(200, 0), P3(240, 150), P4(40, 180):
Cross-product sum 1 = (0×0) + (200×150) + (240×180) + (40×0) = 0 + 30,000 + 43,200 + 0 = 73,200
Cross-product sum 2 = (0×200) + (0×240) + (150×40) + (180×0) = 0 + 0 + 6,000 + 0 = 6,000
Area = ½ × |73,200 − 6,000| = ½ × 67,200 = 33,600 sq ft (0.77135 acres or 3,121.54 m²)
Triangulation & Decomposition for Irregular Polygons
When vertex coordinates are unavailable, any non-self-intersecting n-sided irregular polygon can be divided into n − 2 triangles using internal diagonal chords. Compute the area of each constituent triangle using Heron's formula:
Interactive Polygon Formula & Shoelace Sandbox
Select a mathematical proof mode to visualize how polygon area formulas are derived and calculated:
How to Find the Area of Common Polygons (3 to 12+ Sides)
Standard geometric equations apply to regular and irregular polygons based on the number of boundary sides:
4-Sided Polygon Area (Quadrilaterals & Uneven Lots)
A four-sided polygon (quadrilateral) includes squares, rectangles, parallelograms, trapezoids, rhombuses, and irregular four-sided lots. For a cyclic quadrilateral (where all 4 vertices touch a common circle), Brahmagupta's formula computes area directly from side lengths:
For general irregular 4-sided lots that are non-cyclic, use the coordinate Shoelace method or measure one diagonal to split the lot into two triangles.
5-Sided Polygon Area (Pentagon Calculator: Regular & Irregular)
A 5-sided polygon is a pentagon. For a regular pentagon with equal side length s, the exact area formula is:
Area = 1.7204774 × 50² = 1.7204774 × 2,500 = 4,301.19 sq ft (0.09874 acres or 399.59 m²)
For an irregular 5-sided polygon with differing side lengths, use Coordinates Mode in the calculator to plot the 5 corner points for exact area calculation via the Shoelace algorithm.
6-Sided Polygon Area (Hexagon Calculator)
A regular 6-sided hexagon is composed of 6 congruent equilateral triangles. The area formula is:
Area = 2.5980762 × 60² = 2.5980762 × 3,600 = 9,353.07 sq ft (0.21472 acres or 868.93 m²)
7-Sided Polygon Area (Heptagon)
A 7-sided polygon is a heptagon (or septagon). For a regular heptagon with side length s, the area formula is:
Area = 3.6339124 × 40² = 3.6339124 × 1,600 = 5,814.26 sq ft (0.13348 acres or 540.16 m²)
8-Sided Polygon Area (Octagon Calculator)
An 8-sided polygon is an octagon. For a regular octagon with side length s, the area formula is:
Area = 4.8284271 × 50² = 4.8284271 × 2,500 = 12,071.07 sq ft (0.27711 acres or 1,121.44 m²)
9-Sided and 12-Sided Polygon Area (Nonagon & Dodecagon)
For a regular 9-sided nonagon (enneagon) and 12-sided dodecagon, the algebraic area formulas are:
Dodecagon Area = 3(2 + √3) × s² ≈ 11.1961524 × s²
Area = 11.1961524 × 20² = 11.1961524 × 400 = 4,478.46 sq ft (0.10281 acres or 416.06 m²)
Interactive Common Polygons Evaluator (3 to 12+ Sides)
Select a common polygon classification and adjust side length to see real-time geometry calculations:
Irregular Polygon Area vs Regular Polygons
A regular polygon is both equilateral and equiangular, while an irregular polygon has sides of differing lengths, unequal interior angles, or both. There are three core mathematical differences between regular and irregular polygons:
To measure the area of an irregular land parcel, record each boundary corner using Cartesian coordinates (feet or meters) from a certified plat map, total station survey, or GPS receiver. Enter the ordered coordinates into the polygon area calculator above to compute the exact acreage using the Gauss Shoelace algorithm.
Interactive Irregular Polygon & Parcel Plotter
Select a real-world land boundary scenario to inspect how coordinates determine the exact acreage of irregular parcels:
Polygon Geometry Reference & Interior Angles Table
The interior angles of any planar n-sided polygon sum to (n − 2) × 180°. The following reference table outlines the geometric properties, interior angle measurements, and area multipliers for regular polygons with 3 to 12 sides:
| Polygon Name | Sides (n) | Interior Angle Sum | Interior Angle (Regular) | Area Multiplier (k × s²) | Diagonals |
|---|---|---|---|---|---|
| Triangle (Equilateral) | 3 | 180° | 60.00° | 0.43301 × s² | 0 |
| Quadrilateral (Square) | 4 | 360° | 90.00° | 1.00000 × s² | 2 |
| Pentagon | 5 | 540° | 108.00° | 1.72048 × s² | 5 |
| Hexagon | 6 | 720° | 120.00° | 2.59808 × s² | 9 |
| Heptagon (Septagon) | 7 | 900° | 128.57° | 3.63391 × s² | 14 |
| Octagon | 8 | 1,080° | 135.00° | 4.82843 × s² | 20 |
| Nonagon (Enneagon) | 9 | 1,260° | 140.00° | 6.18182 × s² | 27 |
| Decagon | 10 | 1,440° | 144.00° | 7.69421 × s² | 35 |
| Hendecagon (Undecagon) | 11 | 1,620° | 147.27° | 9.36564 × s² | 44 |
| Dodecagon | 12 | 1,800° | 150.00° | 11.19615 × s² | 54 |
Land Measurement & Area Unit Conversion Reference
The following reference table provides exact mathematical conversion factors between common Imperial and Metric land area units:
| From Unit | To Unit | Mathematical Operation | Exact Conversion Factor |
|---|---|---|---|
| Acres (ac) | Square Feet (sq ft) | Multiply by | 43,560 ft² |
| Acres (ac) | Square Meters (m²) | Multiply by | 4,046.8564224 m² |
| Acres (ac) | Hectares (ha) | Multiply by | 0.40468564224 ha |
| Acres (ac) | Square Yards (sq yd) | Multiply by | 4,840 yd² |
| Acres (ac) | Square Miles (sq mi) | Divide by | 640 |
| Square Feet (sq ft) | Acres (ac) | Divide by | 43,560 |
| Square Meters (m²) | Acres (ac) | Multiply by | 0.00024710538 |
| Hectares (ha) | Acres (ac) | Multiply by | 2.4710538147 |
| Square Miles (sq mi) | Acres (ac) | Multiply by | 640 ac |
| Square Kilometers (km²) | Hectares (ha) | Multiply by | 100 ha |
Common Acreage Fractions Quick Reference
Use this quick conversion chart for evaluating land parcels, residential lots, and agricultural plots:
| Acre Fraction | Acres | Square Feet (sq ft) | Square Meters (m²) | Hectares (ha) |
|---|---|---|---|---|
| 1/10 Acre | 0.10 ac | 4,356 ft² | 404.69 m² | 0.04047 ha |
| 1/4 Acre | 0.25 ac | 10,890 ft² | 1,011.71 m² | 0.10117 ha |
| 1/3 Acre | 0.333 ac | 14,520 ft² | 1,348.95 m² | 0.13490 ha |
| 1/2 Acre | 0.50 ac | 21,780 ft² | 2,023.43 m² | 0.20234 ha |
| 1 Full Acre | 1.00 ac | 43,560 ft² | 4,046.86 m² | 0.40469 ha |
| 2 Acres | 2.00 ac | 87,120 ft² | 8,093.71 m² | 0.80937 ha |
| 5 Acres | 5.00 ac | 217,800 ft² | 20,234.28 m² | 2.02343 ha |
| 10 Acres | 10.00 ac | 435,600 ft² | 40,468.56 m² | 4.04686 ha |
Frequently Asked Questions
Common questions about acres, acreage calculations, land measurement, and unit conversions.
To calculate the area of an irregular polygon without coordinates, divide the polygon into non-overlapping triangles using interior diagonal lines (triangulation method). Measure all three side lengths of each triangle, calculate each individual triangle area using Heron's formula: Area = √(s(s − a)(s − b)(s − c)) where s = (a + b + c) ÷ 2, and sum the areas of all triangles to obtain the total polygon area.
To find the area of a regular polygon using the apothem, multiply half the apothem by the perimeter: Area = ½ × a × P, where a is the apothem (the perpendicular distance from the geometric center to the midpoint of any side) and P is the total perimeter (number of sides n multiplied by side length s, P = n × s). For example, a regular pentagon with side length 10 feet (3.05 meters) and apothem 6.88 feet (2.10 meters) has perimeter P = 50 feet and Area = ½ × 6.88 ft × 50 ft = 172 square feet (15.98 square meters).
The Shoelace formula (also known as Gauss's area formula or the Surveyor's formula) calculates the exact area of any polygon from the Cartesian coordinates (X, Y) of its ordered vertices. The formula is: Area = ½ × |(x₁y₂ + x₂y₃ + ... + xₙy₁) − (y₁x₂ + y₂x₃ + ... + yₙx₁)|. Cross-multiply adjacent coordinates around the perimeter in a consistent clockwise or counterclockwise sequence, subtract the reverse cross-products, take the absolute value, and divide by 2.
Yes, you can calculate the area of a 5-sided polygon with all different side lengths by using vertex coordinates with the Shoelace formula or by splitting the pentagon into 3 triangles using 2 internal diagonals. Side lengths alone without angles or diagonals cannot determine a single fixed area because a non-rigid 5-sided polygon can flex into multiple configurations with different interior angles.
To find the area of a 4-sided irregular lot (quadrilateral), record the four corner boundary points as (X, Y) coordinates and apply the Shoelace formula: Area = ½ × |(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) − (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)|. If coordinates are unavailable, measure at least one diagonal distance across the lot to divide the quadrilateral into 2 triangles, then compute the sum of both triangle areas using Heron's formula.
The area formula for a 6-sided regular hexagon with side length s is: Area = (3√3 ÷ 2) × s² ≈ 2.598076 × s². A regular hexagon consists of 6 congruent equilateral triangles each having area (√3 ÷ 4) × s². For example, a regular hexagon with side length 20 feet (6.10 meters) has an area of 2.598076 × (20 ft)² = 1,039.23 square feet (96.55 square meters or 0.02386 acres).
To calculate polygon area in acres, first compute the total surface area in square feet (sq ft), then divide by 43,560 (the number of square feet in one international acre): Acres = Area in sq ft ÷ 43,560. If the area is measured in square meters (m²), multiply by 0.000247105 or divide by 4,046.8564 to convert directly to acres.
A regular polygon is both equilateral (all side lengths are identical) and equiangular (all interior angles are equal), while an irregular polygon has sides of differing lengths, unequal interior angles, or both. Regular polygons have a single fixed geometric center, equal apothems, and follow standard algebraic area formulas, whereas irregular polygons require coordinate geometry (Shoelace formula) or triangular decomposition for accurate surface area computation.
GPS coordinates work in the polygon area calculator by projecting spherical latitude and longitude coordinates into planar Cartesian (X, Y) coordinates in feet or meters using map projections (such as Universal Transverse Mercator, UTM, or local state plane coordinates). Once projected onto a flat plane, the vertex coordinates are evaluated in sequential order using the Shoelace formula to compute parcel acreage and perimeter.