Subject: Geometry

Area Of Polygons And Circles

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.

Area of Circles

Finding the area of a circle depends on three factors. We can get the area by means of the circle's circumference, radius or diameter - all of which are parts of a circle.

Parts of a circle

  • Circumference - It is the perimeter of the circle or the edge of the circle. In the figure above, the circumference is the black curve enclosing the circle.
  • Radius - Radius is any line connecting the center of the circle to any points on the circle.
  • Diameter - It is a line segment connecting two points on a circle that passes the center. The diameter's length is twice the radius' length.

Finding the area of a circle with known circumference (C).

A = \frac{c^2}{4\pi}

Finding the area of a circle with known radius (r).

A = \pi r^2

Finding the area of a circle with known diameter (d).

A = \frac{\pi d^2}{4}