Understanding Triangles

Triangles are a simple shape, but they can teach us a lot about geometry.

It’s easy enough to draw a triangle – for example, by making three dots on a piece of paper and then drawing straight lines connecting them. But because it’s drawn on a flat piece of paper, this makes it a “polygon” – a shape in a two-dimensional space where all of its inner space is enclosed by unbroken lines.

That seems fairly simple. Even the word “triangle” simply means “three angles”. But the whole point of identifying the types of polygon in geometry is to understand the mathematics behind them – for example, the area of the space they enclose, and relation of that area to the height and width of the triangle’s sides or the degree of its angles.

Here’s where it gets interesting. It’s easy to know the area of a square – just multiply its length by its height. And with just one extra step, the same formula could be used to determine the area of a parallelogram. (Since its two sides are parallel, a parallelogram forms a simple square when its left half is moved to its right side!) But even with triangles, this same formula can be used. If a triangle’s top half is flipped upside-down (and placed against one of its sides), the triangle instantly becomes a parallelogram! And at that point, its area is simply its length times the height of the parallelogram (which we know to be half the height of the original triangle.)

Area = 1 / 2 (base x height)

Another interesting fact about triangles is that its three angles will always add up to 180 degrees. This makes it possible to determine the radius of the third angle if the other two angles are known. And this is especially useful in the case of a right angle, where two perpendicular sides meet in a 90 degree angle. This means that the other two angles will always add up to 90 degrees, which in geometry is called “complementary” angles. And if both the sides touching the right angle are the same length, then the other angles are also going to be equal, which means they’ll be 45 degrees each.

But one of the most important rules for right triangles is the Pythagorean theorem. If you square the length of the hypotenuse – the side opposite the right angle – it will equal the sum of the squared length for the other two sides. The classic example is a right triangle with a height of 4 inches and a width of 3 inches. Pythagorean’s theorem predicts its hypotenuse will be 5 inches long.

(3*3) + (4*4) = (5*5)

Drawing and measuring a few right triangles will prove that Pythagorean’s theorem is indeed correct!