Subject: Geometry

Polygon Perimeter And Area


Natural formation of polygon shapes.


Calculating the perimeter and area of polygons has already been practiced long time ago. Ancient houses, establishments, structures and landmarks are all made up of different forms and types of polygons. To ensure that the complex task of putting together pieces of polygons will be successful, accurate measurement techniques are needed. Today, we are lucky to have technology on our side. Calculating areas and perimeters of different shapes and volumes such as polygons are carried out easily by technology agents such as computers and etc.

In this section, let us understand how to calculate the dimensional properties of polygons. These properties are perimeter and area.

Calculating the Perimeter of Polygons

The perimeter of a polygon is the total distance around the outside of the polygon or the total length of the sides or edges of a polygon. Perimeter of a polygon is determined by adding the lengths of all the sides or edges. For example, a triangle, which is a polygon, whose sides are of dimensions 5cm, 6cm and 9 cm has a perimeter of 20cm.

Perimeter irregular polygons

Sides of irregular polygons are not all equal. So in calculating its perimeter, we have to add individually the lengths of the sides. In general, when an irregular polygon has n-sides (n-gon) and S_1, S_2, S_3,...S_n are the lengths of the sides, then its perimeter P is calculated by:

P =S_1+S_2+S_3+\cdot \cdot \cdot+S_n

Perimeter of regular polygons

In regular polygons, the same formula in finding the perimeter is applied. However, the sides or edges of regular polygons are all equal, the formula above can take another form. Generally,

P =S_1+S_2+S_3+\cdot \cdot \cdot+S_n

but since all sides are equal, then its lengths must also be equal. Thus, S_1= S_2 = S_3 = S_n = S. Then, the form above can be modified into

P =S_1+S_2+S_3+\cdot \cdot \cdot+S_n

=S + S + S \cdot \cdot \cdot S_n
= nS

where n is the number of sides of a regular polygon. So for example, a hexagon has 6 sides and each side is 8m long. Thus its perimeter is 8m*6 = 48m.

Calculating the Area of Polygons

The area of a polygon is a measure of the size of the region bounded by the sides or edges. This tells us how many square units are needed to fill the entire shape of the polygon.Typically, area is measured in square units such as centimeter square (cm^2) or meter square (m^2). Since area is the number of squares needed to fill the entire shape, then area would just be a sum of all the squares that filled the shape. For example, in the figure below there are 18 squares inside the rectangle. So the area of the rectangle below is 18 square units.

If the polygon has given lengths of sides or edges, then to calculate the area, all we have to do is multiply its width with its length. For example, the rectangle below (which is the same rectangle above) has length 6 units and a width of 3 units. Thus its area A is 3*6 =18 square units.

Area of a Triangle

In the examples above, we mainly deal with rectangular area. Now let's shift our attention to triangles. In regular triangles, the area is just half the product of its base and height.

A = \frac{1}{2}base*height

The formula presented above works only for right, isosceles and equilateral triangle. For scalene triangle, we will use the so called Heron's formula.

A = \sqrt{s(s-a)(s-b)(s-c)}

where a, b and c are the corresponding measures of the lengths of the sides of a scalene triangle and s is given by

s=\frac{a+b+c}{2}
.

Area of a Square

In squares, determining its area is simply done by squaring the length of its sides. Thus,

A = s^2

where s is the length of the square's side.

Area of a Parallelogram

Determining the area of a parallelogram is the same as finding the area of a rectangle. To do it, we only multiply the length of the base of the parallelogram times the length of the height.

A = base*height

Area of a Trapezoid

If you look at the trapezoid below, we can deduce two triangles inside the trapezoid. Thus the area of a trapezoid is just the sum of the area of two deduced triangles. We know already the area of triangles thus, the area of a trapezoid is given by

A = \frac{1}{2}base1*height + \frac{1}{2}base2*height
=\frac{1}{2}height*(base1 + base2)

Area of Regular Polygons

In the discussions above, we have already tackled about the area of the first few regular polygons. These are the equilateral triangle and square. In this subsection, we will discuss about the generalized formula for finding the area of a regular polygon. There are many ways to calculate the area of a polygon. The choice of methods to use depends on the known parts of a regular polygon. So before introducing you to the different formula in finding the area of a regular polygon. Let me introduce to you first the different part of a regular polygon.

Parts of regular polygons

  • Center - In the figure above, the center is the red dot. It is a point inside a regular polygon where the distance to all the vertices are equal.
  • Vertex - Vertex is the point in a regular polygon where the sides or edges meet.
  • Radius - The line that connects the vertex and the center is called the radius.
  • Apothem - Apothem is the line from the center to a point in the edges or sides that is located halfway between two consecutive vertices. This line makes a right angle with the sides or edges.

Finding the area with known sides.

The sides or edges of regular polygons are equal. Thus it would be easy to find the its area. The area of a regular polygon according to its sides or edges is given by;

A = \frac{ns^2}{4\tan{\frac{\pi}{n}}}

where;

  • n = the number of sides of the regular polygon
  • s = the length of the regular polygon's sides

Finding the area with known apothem.

If we know the apothem, then the area will follow as;

A = m^2n\tan{\frac{\pi}{n}}

Finding the area with known apothem and sides.

If both apothem and sides or edges are given, then we calculate area of a regular polygon easily as;

A = \frac{AP}{2}

Finding the area with known radius.

Having a known radius can lead us to finding the area of a regular polygon as;

A = \frac{r^2n\tan{\frac{2\pi}{n}}}{2}

Area of irregular polygons

Irregular polygons are very hard to characterize. It can take any form, sizes and finding a general formula for its area is very complex. However, I would like to share with you one of the most effective and most used technique in finding the area of irregular polygons. This technique is called triangulation. This is done by transforming the irregular polygons into triangles. And since the area of triangles are well-known, then the area of irregular polygon would then just be equal to the sum of the individual triangles that comprises the polygon. Triangulation is known to be effective to almost all types of irregular polygons given that the lengths of all the sides or edges are known. For example, look at the figure below.

As seen in the figure above, the polygon is divided into small triangles with area A1, A2, A3, A4, A5 and A6. Then the area of the polygon would be the sum of the area of the triangle.

To find the area of the small triangles, we will use Heron's formula (presented above) since the triangles formed are scalene. But, if you look at A1, we can't directly use Heron's formula since the third side of the triangle (the side connecting A1 and A2) is unknown. So we have to solve first that unknown side using the law of cosines in trigonometry. After that, the same process will be applicable to other small triangles. So basically, a combination of Heron's formula and law of cosines can get this thing done.