Question

What is the formula of taking out the area of a irregular decagon?

Answer 1

Area of an Irregular Polygon

Unlike a regular polygon, unless you know the coordinates of the vertices, there is no easy formula for the area of an irregular polygon. Each side could be a different length, and each interior angle could be different. It could also be either convex or concave.

If you know the coordinates of the vertices of the polygon, there are two methods:

A manual method. See Area of a polygon (Coordinate geometry).A computer algorithm. See Algorithm to find the area of any polygon

So how to do it?

One approach is to break the shape up into pieces that you can solve - usually triangles, since there are many ways to calculate the area of triangles. Exactly how you do it depends on what you are given to start. Since this is highly variable there is no easy rule for how to do it. The examples below give you some basic approaches to try.

1. Break into triangles, then add

In the figure on the right, the polygon can be broken up into triangles by drawing all the diagonals from one of the vertices. If you know enough sides and angles to find the area of each, then you can simply add them up to find the total. Do not be afraid to draw extra lines anywhere if they will help find shapes you can solve. 

Here, the irregular hexagon is divided in to 4 triangles by the addition of the red lines. ( See Area of a Triangle)

2. Find 'missing' triangles, then subtract

In the figure on the left, the overall shape is a regular hexagon, but there is a triangular piece missing. 

We know how to find the area of a regular polygon so we just subtract the area of the 'missing' triangle created by drawing the red line. (See Area of a Regular Polygon and Area of a Triangle.)

3. Consider other shapes

In the figure on the right, the shape is an irregular hexagon, but it has a symmetry that lets us break it into two parallelograms by drawing the red dotted line. (assuming of course that the lines that look parallel really are!) 

We know how to find the area of a parallelogram so we just find the area of each one and add them together. (See Area of a Parallelogram).

As you can see, there an infinite number of ways to break down the shape into pieces that are easier to manage. You then add or subtract the areas of the pieces. Exactly how you do it comes down to personal preference and what you are given to start.

4. If you know the coordinates of the vertices

If you know the x,y coordinates of the vertices (corners) of the shape, there is a method for finding the area directly. See Area of a polygon (Coordinate geometry). This works for all polygon types (regular, irregular, convex, concave). There is also a computer algorithm that does the same. See Algorithm to find the area of any polygon

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref

Other polygon topics

General

Polygon general definitionQuadrilateralRegular polygonIrregular polygonConvex polygonsConcave polygonsPolygon diagonalsPolygon trianglesApothem of a regular polygonPolygon centerRadius of a regular polygonIncircle of a regular polygonIncenter of a regular polygonCircumcircle of a polygonParallelogram inscribed in a quadrilateral

Types of polygon

SquareDiagonals of a squareRectangleDiagonals of a rectangleGolden rectangleParallelogramRhombusTrapezoidTrapezoid medianTrapeziumKiteInscribed (cyclic) quadrilateralInscribed quadrilateral interior anglesInscribed quadrilateral areaInscribed quadrilateral diagonals

Area of various polygon types

Regular polygon areaIrregular polygon areaRhombus areaKite areaRectangle areaArea of a squareTrapezoid areaParallelogram area

Perimeter of various polygon types

Perimeter of a polygon (regular and irregular)Perimeter of a trianglePerimeter of a rectanglePerimeter of a squarePerimeter of a parallelogramPerimeter of a rhombusPerimeter of a trapezoidPerimeter of a kite

Angles associated with polygons

Exterior angles of a polygonInterior angles of a polygonRelationship of interior/exterior anglesPolygon central angle

Named polygons

Tetragon, 4 sidesPentagon, 5 sidesHexagon, 6 sidesHeptagon, 7 sidesOctagon, 8 sidesNonagon Enneagon, 9 sidesDecagon, 10 sidesUndecagon, 11 sidesDodecagon, 12 sides