Chapter 5 Geometry

Section 5.4 Polygons, Area and Circumference

5.4.5 Area

The area of a surface equals the number of unit squares required to cover this surface completely.

Let us first consider rectangles. If the sides of the rectangle are of lengths $a$ and $b$, then the rectangle contains $b$ rows with $a$ unit squares, i.e. $b·a$ unit squares.

With this, the area of a right triangle can be calculated easily. Let $ABC$ be a right triangle rotated by an angle of $180{}^{\circ }$. If the original and the rotated triangle are merged along the hypotenuse, one obtains a rectangle.

The area of the right triangle is then half the area of the rectangle, i.e. $F=\frac{1}{2}·a·b$.
And how is the area calculated if the triangle is not right-angled?
Every triangle can be divided into two right triangles by drawing a line from one vertex to the opposite side such that this line is perpendicular to the side. This line is called the altitude $h_i$ of a triangle on a specific side $i$, where $i$ is the index of the side $a$, $b$, or $c$.
Depending on whether the new line is interior or exterior to the triangle, the area of the triangle equals the sum or the difference of the areas of the two resulting right triangles:

Thus, on the left, we have (if $F_{\Delta }$ is the area of the triangle $\Delta$)

${\displaystyle F_{ABC}=F_{DBC}+F_{ADC}=\frac{1}{2}·h_c·c_2+\frac{1}{2}·h_c·c_1=\frac{1}{2}·h_c·(c_2+c_1)=\frac{1}{2}·h_c·c\hspace{0.5em}.}$
On the right, we have

${\displaystyle F_{UVW}=F_{XVW}-F_{XUW}=\frac{1}{2}·h_w·w_2-\frac{1}{2}·h_w·w_1=\frac{1}{2}·h_w·(w_2-w_1)=\frac{1}{2}·h_w·w\hspace{0.5em}.}$
Thus the area can always be calculated from the length of one side and the length of the altitude perpendicular to the corresponding side.



Using the formula for the area of triangles, areas of polygons can also be calculated. This is due to the fact that every polygon can be divided into triangles by adding diagonals to the polygon until all subareas are triangles. However, the considerations will remain restricted here to a few simple shapes. In the following example, the polygon can be divided into a triangle and a rectangle. As a result, the calculation will be particularly easy.


Finally, we will calculate the area of a circle. Info Box 5.2.6 introduced the number $\pi$ describing the ratio of the circumference of a circle to its radius. The formula for the area of the circle also involves $\pi$.