- How to find the area of a pentagon?
- Area of a regular pentagon
- Area of an irregular pentagon
- Gaussian determinant
- References
The area of a pentagon is calculated using a method known as triangulation, which can be applied to any polygon. This method consists of dividing the pentagon into several triangles.
After this, the area of each triangle is calculated and finally all the areas found are added. The result will be the area of the pentagon.
The pentagon could also be divided into other geometric shapes, such as a trapezoid and a triangle, such as the figure on the right.
The problem is that the length of the greater base and the height of the trapezoid are not easy to calculate. Also, the height of the red triangle must be calculated.
How to find the area of a pentagon?
The general method for calculating the area of a pentagon is triangulation, but the method can be straightforward or a bit longer depending on whether the pentagon is regular or not.
Area of a regular pentagon
Before calculating the area it is necessary to know what the apothem is.
The apothem of a regular pentagon (regular polygon) is the smallest distance from the center of the pentagon (polygon) to the midpoint of one side of the pentagon (polygon).
In other words, the apothem is the length of the line segment that goes from the center of the pentagon to the midpoint of one side.
Let us consider a regular pentagon such that the length of its sides is "L". To calculate its apothem, first divide the central angle α by the number of sides, that is, α = 360º / 5 = 72º.
Now, using the trigonometric ratios, the length of the apothem is calculated as shown in the following image.
Therefore, the apothem has a length of L / 2tan (36º) = L / 1.45.
By triangulating the pentagon, a figure like the one below will be obtained.
All 5 triangles have the same area (for being a regular pentagon). Therefore the area of the pentagon is 5 times the area of a triangle. That is: area of a pentagon = 5 * (L * ap / 2).
Substituting the value of the apothem, we obtain that the area is A = 1.72 * L².
Therefore, to calculate the area of a regular pentagon, you only need to know the length of one side.
Area of an irregular pentagon
We start from an irregular pentagon, such that the lengths of its sides are L1, L2, L3, L4 and L5. In this case, the apothem cannot be used as used before.
After doing the triangulation, a figure like the following is obtained:
Now we proceed to draw and calculate the heights of these 5 interior triangles.
So the areas of the interior triangles are T1 = L1 * h1 / 2, T2 = L2 * h2 / 2, T3 = L3 * h3 / 2, T4 = L4 * h4 / 2, and T5 = L5 * h5 / 2.
The values for h1, h2, h3, h4, and h5 are the heights of each triangle, respectively.
Finally the area of the pentagon is the sum of these 5 areas. That is, A = T1 + T2 + T3 + T4 + T5.
As you can see, calculating the area of an irregular pentagon is more complex than calculating the area of a regular pentagon.
Gaussian determinant
There is also another method by which the area of any irregular polygon can be calculated, known as the Gaussian determinant.
This method consists of drawing the polygon on the Cartesian plane, then the coordinates of each vertex are calculated.
The vertices are enumerated counterclockwise and finally certain determinants are calculated to finally obtain the area of the polygon in question.
References
- Alexander, DC, & Koeberlein, GM (2014). Elementary Geometry for College Students. Cengage Learning.
- Arthur Goodman, LH (1996). Algebra and trigonometry with analytical geometry. Pearson Education.
- Lofret, EH (2002). The book of tables and formulas / The book of multiplication tables and formulas. Imaginative.
- Palmer, CI, & Bibb, SF (1979). Practical mathematics: arithmetic, algebra, geometry, trigonometry and slide rule (reprint ed.). Reverte.
- Posamentier, AS, & Bannister, RL (2014). Geometry, Its Elements and Structure: Second Edition. Courier Corporation.
- Quintero, AH, & Costas, N. (1994). Geometry. The Editorial, UPR.
- Ruiz, Á., & Barrantes, H. (2006). Geometries. Editorial Tecnologica de CR.
- Torah, FB (2013). Maths. 1st didactic unit 1st ESO, Volume 1. Editorial Club Universitario.
- Víquez, M., Arias, R., & Araya, J. (sf). Mathematics (sixth year). EUNED.