Probably the easiest way to do this requires that you know only two things about your triangle: the length of its base, and its height. With those numbers, you simply insert them into this equation and solve for the area:

Pretty easy, right? Do you understand why this equation works, though? Take a look at the equation again. You might recognize that it looks familiar, almost like the equation to solve the area of a rectangle – only this time, multiply by a half. So the area of a triangle is equal to half of the area of a rectangle. You can very easily see this simply by drawing any old rectangle, label the sides as “base” and “height”, and then go ahead and draw in a diagonal line that connects two of the corners. You’ve just created two equal triangles, and each triangle is half of the original rectangle! Well, that’s great if you have a triangle that has one side perfectly horizontal, and the other 90 degrees up perfectly vertical. That is literally half of a rectangle. But what about triangles that have more irregular angles?

Like a triangle with 31, 82, and 67 degrees? No right angles there, and it certainly doesn’t look like half of a rectangle. Check this out though. It still works! Orient your triangle so that it’s long side is on the bottom. Then, draw in a line from its highest point straight down to intersect the bottom.

So far, we have just identified its base and its height. Now, for the fun part! Separate your triangle on the vertical line you just drew, and then take your two new smaller triangles and switch sides – put the one on the left over on the right now, but right next to each other, touching at their corner. Now we have a weird structure that has the same base length, and has two vertical walls that have the same height. Now if you just draw in another line to connect the two tops, you can create a rectangle. And, if you were to measure the area of the new triangle created when you connected the tops, you would find that its area is exactly equal to the area of your first triangle, and obviously together they form the complete area of the rectangle.

So, there you have it. Visual proof of the triangle area formula! That’s a fantastic method to use to find the area of the triangle if you know its base and height (or “altitude” as some may call it). But what about if you don’t know its height? The height of right angle triangles is easy because it is equal to one of the sides. But for other triangles, if you know the side lengths, that doesn’t always mean you can immediately determine the height. In that case, there is another formula that you can use, called Heron’s (or Hero’s) formula. You can use Heron’s formula to calculate the area of a triangle where you know the lengths of all of the sides.

Heron’s formula is sometimes referred to as the “irregular triangle calculation method,” and is named after Heron (or Hero) of Alexandria. Heron was an engineer and mathematician in Ancient Greece, who is credited with inventing many thing, including an early steam engine , a windwheel, and a vending machine. (Check out this Wikipedia page for more information on some of his inventions and accomplishments!) His formula for calculating the area of a triangle contains a few steps, but thankfully, they are not all that difficult. The first thing you have to do is to calculate the half-perimeter (or the semi-perimeter, s). You do this by simply summing the three side lengths (a, b, and c) and dividing in half. Once you’ve done that, then you just plug numbers into the following equation:

It might be easier to remember this equation by breaking it down and looking at what each part is. If you consider that the three bracketed terms are really just the difference in side length from the half-perimeter, then all you need to remember is to multiply those three values by the half-perimeter itself, and then square root the whole thing. I know that sounds like a mouthful, but having some degree of understanding about what more complex formulas are based on is always helpful.

Somewhat related to this method is a much simplified version, which is specifically applicable to equilateral triangles. (Of course, we all know that equilateral triangles are special triangles whose sides are equal and all angles are 60 degrees.) It’s a much simpler equation to remember, without needing to make a first calculation like with Heron’s formula. Here is the formula for the area of an equilateral triangle:

The final method that I will describe here uses trigonometry to find the area. You can use this formula on any triangle, provided that you know the length of two of the sides and the angle between them. (Note: there are variations of this formula, depending on what sides and angle are known). It’s fairly easy to remember, as long as you just think about ABC! Here’s the formula:

This goes back to my diagram above, with the inscribed “height” line in the triangle. Using a basic trig identity for sine, you can calculate the height using the hypotenuse of one of the smaller triangles and a known angle. Then, you simply take this trig formula and substitute it in for the height term in our original triangle area formula (1/2 base x height). With some simple rearranging, you can come up with our final half absinc formula here!

That’s all the methods that I’m going to discuss here, though you now know four ways of calculating the area of a triangle! Just pick the right methods for your specific problem, and you will only be a few steps away from solving the area!

The definition of the Theorem of Pythagoras states that the square of the hypotenuse is the sum of the squares of the other two sides. That’s the wordy version, but it’s all you need to remember how to use it. Personally, I can almost recall this phrase even more easily than the mathematical expression that defines it, and that is a very simple equation to remember! This is how quick and painless the formula is:

 c² = a² + b²

That is all there is to it! This is known as the Pythagorean Equation. It’s amazing that such a small equation can have such a wide-reaching application. This expression says exactly what I defined above, as it provides a simple relationship connecting the three sides of a triangle. The only condition required is that we have a right angle triangle, with c in the expression being the length of the hypotenuse. So then we have the square of the hypotenuse (c) equal to the sum of the squares of the other two sides. If you think about this relationship, you can also deduce that the hypotenuse will always be the longest side, and always less than the sum of the other two.

Now you might be wondering just WHY this equation is true. Perhaps you want proof. OK then, here is a diagram to prove this to you. What I have done is created a right triangle with sides a = 3, b = 4, and c = 5, and then built a square off of each of the triangle’s sides. (Get it? I squared each side!) If you do the math, you will find that the sum of the area of the side square and bottom square (a and b) are equal to the area of the square built off the hypotenuse. Visually, this is all that the math means.

Hopefully, the diagram above is sufficient enough to demonstrate this concept. However, maybe you would prefer to see video proof that a² + b² = c²? Here is a fantastic video that I found that perfectly demonstrates this theorem. It is a wonderful, watertight demonstration of the theorem of Pythagoras that shouldn’t leave you with any doubt.

Here’s another page with a thought exercise on it to prove the theorem. This Pythagorean card table problem is another good demonstration of putting these mathematics concepts to good use!

This is one of the most basic trig concepts, and is probably one of the first concepts that were taught in trigonometry. In fact, there are several different variations of proofs that have been developed over the years, based on concepts of proportionality (similar triangles), geometry (Euclid’s proof, or rearrangement), or algebra and calculus. It is very important and very useful, as you will undoubtedly be able to make use of it in a wide range of solutions to math questions for years.

c² = a² + b² – 2ab cos C

Upon a quick analysis, you can see that this law requires two of the three sides in a triangle, as well as the angle opposite the unknown side (or the angle contained by both of the known sides) to provide you with the unknown third side. Here is another trick to think of it: use your thumb and index finger to form a V shape that will represent your triangle, and if you know the lengths of the two sides (fingers) and the angle between them, you can find the remaining side by using the Cosine Law. This is useful for finding the third side of any triangle when two sides and the angle between them are known.

Let’s try an example, where we will solve for the unknown sides and angles.

First, assess the information that you are given and come up with a strategy to solve this triangle. (The phrase “solve the triangle” is often used to instruct you to determine all of the missing angles and sides). In this case, there are two sides and the angle between them known – the perfect case for the Cosine Law to deduce the unknown side! Once we’ve gone that far, we will have then obtained a complete ratio (side and angle) to use in the Sine Law to solve for the remaining two angles. So, now let’s put our strategy to work!

So by the Cosine Law:

c² = 6² + 8² – 2(6)(8) cos 60

c² = 36 + 64 + 96 cos 60

c² = 36 + 64 – 48

c² = 52

c = 7.2

Now that we have done that, we have obtained the final side length, and therefore a complete side/angle ratio to use with the Sine Law! And now, you can find the rest of the angles to fully describe the triangle!

and

I’ll leave those for you to solve. But that’s it! Using a combination of the Cosine Law and Sine Law, you can completely solve any triangle that you are given. These are extremely powerful and useful equations!

Also, on a side note… as I mentioned at the start of this post, the Cosine Law is a generalization of the Pythagorean Theorem, which specifically applies to right angle triangles. You can see that if you are working with a right angle triangle and substitute in 90 degrees to the Cosine Law, it reduces down to the Pythagorean theorem:

c² = a² + b² – 2ab cos C

but

cos 90 = 0

so

2ab cos (90) = 0

reduces to 0, and so

 c² = a² + b²

The Pythagorean Theorem!

Let’s consider a triangle that has three different angles, none of which are right angles. The standard naming scheme still applies, although now there is no hypotenuse (remember, the hypotenuse is opposite to the 90 degree angle).

So, now we have our triangle, what is sine law? This trigonometry sine law says that the ratio between the sine of an angle and the side opposite to it will be equal for all three angles. In other words: