area of the triangle
Kapitoly: Triangle, The height of a triangle, The weight of a triangle, Circles in a triangle, Right triangle, How to draw a triangle, Area of the triangle, The Pythagorean Theorem
We can find the area of a triangle in two main ways. Either we add the triangle to the parallelogram and calculate the area of that parallelogram or we use Heron's formula. You can also watch the article as a video on YouTube!
Completing on a parallelogram
First, let's see how we calculated the area of a triangle when it was a right triangle. We added the triangle to the rectangle, calculated the area of the rectangle and divided the result by two. This is shown graphically in the following figure:
But we can't easily add a triangle that is not a right triangle to a rectangle. But we can add it to a parallelogram. So we have the following triangle ABC:
We will complete this triangle to a parallelogram by drawing a line from the point C which will be parallel to the line AB and will also be the same length. Then, we will draw a line from the point A so that it is parallel to the line BC and is the same length. We get this parallelogram:
The question is how do we calculate the area of this parallelogram. We cannot multiply two adjacent sides as in the case of a rectangle. However, we can easily make a rectangle out of this parallelogram which will have the same content as the parallelogram. The following triangles highlighted in red have the same content:
If we move the triangle AED in place of the triangle BFC, we get a rectangle with the same content:
This is already a rectangle and its content is thus equal to the product of the lengths of the two adjacent sides. But what is the length of the side AE or BF? If you look carefully at the original triangle ABC, highlighted in bold, you will see that the length of the side AE exactly matches the length of the height of the triangle from the vertex C. So what can we write about the area of this rectangle? That the content is equal to:
$$S_{\square}=|v_c|\cdot|AB|.$$
where vc is the height from the vertex C. The area of the triangle is then equal to half of that content:
$$S_{\triangle}=\frac{|v_c|\cdot|AB|}{2}.$$
The final formula
Of course, we can generalize the formula from the last chapter to any of the three sides:
$$\begin{eqnarray} S_{\triangle}&=&\frac{|v_a|\cdot|a|}{2},\\ S_{\triangle}&=&\frac{|v_b|\cdot|b|}{2},\\ S_{\triangle}&=&\frac{|v_c|\cdot|c|}{2}. \end{eqnarray}$$
Where va, vb and vc denote the heights to the sides a, b and c.
Heron's formula
If you don't know the length of any height in a triangle, but you know the lengths of all the sides, you can calculate the contents using Heron's formula. It is true that:
$$S_{\triangle}=\sqrt{s\cdot(s-a)\cdot(s-b)\cdot(s-c)},$$
where
$$s=\frac{a+b+c}{2}.$$