Find The Area Of Polygon Calculator

Easily calculate the area of any simple polygon given its vertex coordinates using the Shoelace/Surveyor's formula. Our Area of Polygon Calculator is fast and accurate.

Calculate Polygon Area

Vertex	X	Y	xi * yi+1	yi * xi+1

Table of entered coordinates and intermediate Shoelace formula products.

What is an Area of Polygon Calculator?

An Area of Polygon Calculator is a digital tool designed to determine the surface area enclosed by a simple polygon, given the coordinates (x, y) of its vertices. It typically uses the Shoelace formula (also known as the Surveyor's formula or Gauss's area formula) to compute the area based on the vertex coordinates listed in either clockwise or counter-clockwise order.

This calculator is useful for students, engineers, surveyors, architects, and anyone needing to find the area of an irregular plot of land or any polygonal shape defined by points on a Cartesian plane. It automates a calculation that can be tedious and prone to error when done manually, especially for polygons with many vertices.

Common misconceptions include thinking it can find the area of self-intersecting polygons (the formula gives a different result then) or that it measures perimeter; this tool specifically calculates the area *inside* the polygon.

Area of Polygon Formula and Mathematical Explanation

The most common method for calculating the area of a polygon given its vertices (x₁, y₁), (x₂, y₂), …, (x_n, y_n) is the Shoelace formula. The formula is given by:

Area = 0.5 * | (x₁y₂ + x₂y₃ + … + x_n-1y_n + x_ny₁) – (y₁x₂ + y₂x₃ + … + y_n-1x_n + y_nx₁) |

Or, in summation notation:

Area = 0.5 * | Σⁿ_i=1 (x_iy_i+1) – Σⁿ_i=1 (y_ix_i+1) |

Where (x_n+1, y_n+1) = (x₁, y₁). You list the coordinates in order around the polygon (either clockwise or counter-clockwise), multiply diagonally, sum these products, take the difference of the sums, and then half the absolute value of this difference.

Variables Table

Variable	Meaning	Unit	Typical Range
(xi, yi)	Coordinates of the i-th vertex	Length units (e.g., meters, feet)	Any real number
n	Number of vertices	Dimensionless	3 or more
Area	Area enclosed by the polygon	Square length units (e.g., m^2, ft^2)	Positive real number

Practical Examples (Real-World Use Cases)

Example 1: Area of a Rectangular Plot

Suppose a surveyor measures a rectangular plot of land with vertices at (10, 10), (50, 10), (50, 40), and (10, 40) meters.

Inputs:

• Number of vertices: 4
• Vertex 1: (10, 10)
• Vertex 2: (50, 10)
• Vertex 3: (50, 40)
• Vertex 4: (10, 40)

Calculation:

Sum 1 = (10*10) + (50*40) + (50*40) + (10*10) = 100 + 2000 + 2000 + 100 = 4200

Sum 2 = (10*50) + (10*50) + (40*10) + (40*10) = 500 + 500 + 400 + 400 = 1800

Area = 0.5 * |4200 – 1800| = 0.5 * 2400 = 1200 square meters.

Using the Area of Polygon Calculator with these inputs would give an area of 1200 m².

Example 2: Area of an Irregular Shape

An architect is designing a room with an irregular pentagonal shape with vertices at (0, 0), (5, 1), (6, 4), (2, 5), and (0, 3) feet.

Inputs:

• Number of vertices: 5
• Vertex 1: (0, 0)
• Vertex 2: (5, 1)
• Vertex 3: (6, 4)
• Vertex 4: (2, 5)
• Vertex 5: (0, 3)

Calculation:

Sum 1 = (0*1) + (5*4) + (6*5) + (2*3) + (0*0) = 0 + 20 + 30 + 6 + 0 = 56

Sum 2 = (0*5) + (1*6) + (4*2) + (5*0) + (3*0) = 0 + 6 + 8 + 0 + 0 = 14

Area = 0.5 * |56 – 14| = 0.5 * 42 = 21 square feet.

The Area of Polygon Calculator would quickly provide 21 sq ft.

How to Use This Area of Polygon Calculator

1. Enter Number of Vertices: Start by entering the total number of vertices (corners) your polygon has in the "Number of Vertices" field. The minimum is 3 (a triangle), and our calculator supports up to 12.
2. Input Coordinates: Based on the number of vertices you entered, input fields for the x and y coordinates of each vertex will appear. Enter the x and y values for each vertex in order, either clockwise or counter-clockwise around the polygon.
3. Calculate: Click the "Calculate Area" button.
4. View Results: The calculator will display the total area, the two intermediate sums from the Shoelace formula, and a brief explanation of the formula used.
5. See Table & Chart: A table showing your entered coordinates and intermediate products, along with a visual representation of your polygon, will also be displayed.
6. Reset/Copy: You can use the "Reset" button to clear all inputs or "Copy Results" to copy the calculated area and sums.

The primary result is the area of the polygon in the square units corresponding to the units of your input coordinates. If you entered coordinates in meters, the area will be in square meters.

Key Factors That Affect Area of Polygon Calculator Results

• Accuracy of Coordinates: The precision of the area calculation is directly dependent on the accuracy of the x and y coordinates provided for each vertex. Small errors in coordinates can lead to inaccuracies in the calculated area, especially for small polygons.
• Number of Vertices: While the formula works for any number of vertices (3 or more), more vertices generally mean more complex shapes and more data entry, increasing the chance of input error.
• Order of Vertices: The vertices must be entered in sequential order as you "walk around" the perimeter of the polygon (either clockwise or counter-clockwise). A jumbled order will result in an incorrect area or a self-intersecting polygon calculation.
• Units Used: The units of the calculated area will be the square of the units used for the coordinates. If coordinates are in feet, the area is in square feet. Consistency is key.
• Simple vs. Self-Intersecting Polygons: The Shoelace formula is designed for simple (non-self-intersecting) polygons. If the edges cross, the formula yields a result related to the signed areas of the enclosed regions, which might not be the simple geometric area you expect.
• Co-linear Vertices: Having multiple vertices in a straight line is acceptable, but be precise with their coordinates.

Frequently Asked Questions (FAQ)

1. What is the minimum number of vertices I can enter?

You need at least 3 vertices to form a polygon (a triangle).

2. What happens if I enter the vertices in the wrong order?

If you enter them in clockwise vs. counter-clockwise order, the intermediate sums will swap and the sign before taking the absolute value will flip, but the final area (absolute value) will be the same. However, if you jumble the order completely, you'll get the area of a different, possibly self-intersecting, polygon defined by that order.

3. Can I use this calculator for a 3D shape?

No, this Area of Polygon Calculator is for 2D polygons defined by (x, y) coordinates on a plane.

4. What units should I use for the coordinates?

You can use any consistent unit of length (meters, feet, inches, cm, etc.). The area will be in the square of that unit.

5. Does the calculator handle self-intersecting polygons?

The Shoelace formula will produce a number even for self-intersecting polygons, but it represents a combination of signed areas of the regions formed, not necessarily the simple sum of the areas of the enclosed parts. Our calculator assumes a simple polygon.

6. How accurate is the Area of Polygon Calculator?

The calculation itself is exact based on the Shoelace formula. The accuracy of the result depends entirely on the accuracy of the input coordinates.

7. Can I calculate the area of a circle using this?

No, a circle is not a polygon (it has no straight sides or vertices in the same way). You would approximate a circle with a polygon having many vertices, but it's better to use the formula Area = πr² for a circle. Our circle area calculator might be helpful.

8. What if some of my coordinates are negative?

That's perfectly fine. The Shoelace formula works correctly with negative coordinates, as long as they represent the vertices of your polygon in the Cartesian plane.

Related Tools and Internal Resources

• Circle Area Calculator: Calculate the area of a circle given its radius or diameter.
• Triangle Area Calculator: Find the area of a triangle using various formulas (base-height, Heron's, etc.).
• Rectangle Area Calculator: Quickly find the area of a rectangle.
• Square Footage Calculator: Calculate the area of rooms or land in square feet.
• Coordinate Geometry Basics: Learn more about coordinates and their use in geometry.
• Understanding Geometric Shapes: An overview of different geometric shapes and their properties.