Easily find the area of any irregular five-sided polygon. Simply input the (X, Y) coordinates of the vertices in order to calculate the precise surface area using the coordinate geometry method.
Irregular Pentagon Area Calculator
Irregular Pentagon Area Formula
The most reliable way to calculate the area of an irregular pentagon is using the Shoelace Formula (also known as Gauss’s Area Formula):
Source: Wolfram MathWorld – Polygon Area | Reference: Wikipedia Shoelace Method
Variables:
- (x1, y1) to (x5, y5): The Cartesian coordinates of the five vertices.
- Vertices Order: Points must be listed consecutively around the perimeter (clockwise or counter-clockwise).
What is an Irregular Pentagon Area Calculator?
An irregular pentagon is a five-sided polygon where side lengths and interior angles are not all equal. Unlike a regular pentagon, there is no simple side-based formula like $Area = \frac{1}{4}\sqrt{5(5+2\sqrt{5})}s^2$.
This calculator solves the complexity by using coordinates. This method is preferred in architecture, land surveying, and digital design because it works for any shape regardless of its symmetry or concave/convex nature.
How to Calculate Irregular Pentagon Area (Example)
Assume a pentagon with coordinates: A(0,0), B(4,0), C(5,3), D(2,5), E(-1,2).
- List coordinates: (0,0), (4,0), (5,3), (2,5), (-1,2), and back to (0,0).
- Multiply X of one point by Y of the next: (0*0) + (4*3) + (5*5) + (2*2) + (-1*0) = 41.
- Multiply Y of one point by X of the next: (0*4) + (0*5) + (3*2) + (5*-1) + (2*0) = 1.
- Subtract the results: 41 – 1 = 40.
- Divide by 2: 40 / 2 = 20 square units.
Related Calculators
- Quadrilateral Area Calculator
- Irregular Hexagon Area Tool
- Coordinate Geometry Solver
- Land Area Measurement Tool
Frequently Asked Questions (FAQ)
Can I calculate the area using only side lengths? No. Side lengths alone do not uniquely define the area of an irregular pentagon. You also need either the angles or the coordinates of the vertices.
What if my pentagon is concave? The Shoelace formula used in this calculator correctly handles concave polygons as long as the boundary does not cross itself.
Do the units matter? No, as long as you are consistent. If your coordinates are in meters, the result will be in square meters.
Why is my result negative? If the vertices are entered in clockwise order, the internal math might result in a negative number. This calculator uses the absolute value to ensure a positive area result.