Triangles have some interesting geometric properties that other shapes do not have. To start with, we can define the area of a triangle and see how this relates to its geometric properties (Eq. 1).

Eq. 1: \(a = {hw\over 2}\)

The area of a triangle is defined by the formula (Eq. 1) where a is the area of the triangle, w is the width of the triangle, and h is the height of the triangle. This formula makes sense when you look at some of the geometric properties of triangles, as we will see. It is important to note that the width of a triangle is defined as the length of a single side, and the height of the triangle is defined as the length of a perpendicular line drawn from that side to the opposite corner (Figure 1). In the case of a right triangle, the height is also the length of a second side perpendicular to the first side, but this is not the case with other forms of triangle.

Figure 1: The area of a triangle is defined by the width, multiplied by the height, divided by 2. The width is defined as the length of one of the sides of the triangle, whereas the height is defined as the length of a line perpendicular to the line used to defined the width. This is distinct from the width being the length of a second side of the triangle, or the height and width being a box drawn around the triangle.

If you bisect a triangle using a line from a corner of a triangle to the midpoint of the opposite side of the triangle, you will yield two triangles, both of which have equal areas. The areas of these triangles are each equal to 1/2 of the area of the original triangle (Figure 2). This is apparent, as you have divided the width term in the triangle area equation by 1/2, thus also divided the area by 1/2.

Figure 2: Splitting a triangle along a line from one corner to the midpoint of the opposite side yields two triangles of equal sizes, whose area adds up to the area of the original triangle. This is apparent, from the fact that one of dimensions (the width) is reduced by half, and thus the area of the triangle will also be halved.

If you draw perpendicular line from a corner of a triangle to the opposite side of the triangle (i.e. along the line used to measure the height of the triangle) you can make some interesting observations. Splitting a triangle along the perpendicular line used to define the height yields two right triangles (Figure 3). These triangles do not have equal areas to one another, but their areas do of course add up to equal the original area of the triangle. Also this bisection has some interesting properties

Figure 3: Splitting the triangle along a perpendicular line from a corner to the opposite side yields two right triangles. This means the laws of SOHCAHTOA can be applied to solve the height of a triangle, if the length of sides and the angles of corners are known, by using /(o = h /times /sin o/) where the hypotenuse is the length of one of the sides of the original triangle, and the angle is the known angle in the original triangle, and the height is the length of the side opposite this known angle.

As the right triangles follow the rules of SOHCAHTOA, we can use these to find the height of the triangle, if given the length of the sides of the triangle and the angles of their corners. To do this, we can use the rules of SOH (Eq. 2) and (Eq. 3), finding the opposite side of the right triangle (which is equivalent to the height of the original triangle), using the length of the hypotenuse (which is the length of a side of the original triangle), and the angle of the corner of the triangle is the angle of the corner of our resulting right triangle.

Eq. 2: \(\sin\theta = {o\over h}\)

Eq. 3: \(o=\sin\theta \times h\)

Another property we can use this line for is that we can prove that the area of a triangle is as expressed in (Eq. 1). By producing a rectangle with equal area to the triangle (Figure 4). By splitting our bisecting line halfway, and drawing parallel lines from these until the sides of the triangle are met. Drawing perpendicular lines from these points yields on each side of the original bisector a rectangle and two triangles. These can be rearranged to make two distinct rectangles of equal area to the original triangle.

Figure 4: By splitting a triangle along a perpendicular line, we can yield two right triangles. If we draw perpendicular lines out from the midpoint of this line, these will reach the other two sides at their midpoints. Drawing lines down from these points will reach the midpoints at the remaining two sides. This yields two rectangles and four triangles, which when these shapes are rearranged, they can make two different rectangles of the same area as the original triangle.

These interesting geometric properties are part of what makes triangles such interesting shapes.