What's the area of a triangle? Half its base times its height, right? Indeed, that's the formula. But why? Keep on reading to learn the secret behind this famous formula for the area of a triangle.
Odds are good that at one point or another in your life, you've memorized the formula for the area of a triangle. You know, the classic: Area = 1/2 x base x height.
But the truth is that you didn't really need to memorize that formula. Not because it's not a useful formula to know—triangles show up all over the place and knowing how to calculate their area is extremely useful—but because you already knew how to do it…you just didn't know it.
What exactly am I talking about? How could you have known something that you hadn't yet learned? Stay tuned because that's exactly what we'll be talking about today.
The Triangle Area Formula
The area of a triangle is equal to half the length of the triangle's base times its height. No matter what the triangle looks like, how big or small it is, always and forever.
The base of the triangle is simply any one of its three sides—you get to choose.
What do we mean by base and height? The base of the triangle is simply any one of its three sides—for most triangles any side will do, so you get to choose. Once you've chosen a side for the base, you need to draw a line perpendicular to it that intersects the vertex where the other two sides of the triangle meet. If doing that for the base you’ve chosen on your particular triangle is impossible, you need to try another side for the base. The length of this line is the height of the triangle.
Makes sense, right?
If you draw a few triangles and stare at them for awhile, you'll see that no matter which of the three sides you call the base, the line perpendicular to the base extending to the opposite vertex is always the corresponding height of the triangle. And it turns out that the area of the triangle is always half the length of its base times that height.
But why is that?