Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides. Created by Sal Khan.
I think it's pretty common knowledge how to find the area of the triangle if we know the length of its base and its height. So, for example, if that's my triangle, and this length right here-- this base-- is of length b and the height right here is of length h, it's pretty common knowledge that the area of this triangle is going to be equal to 1/2 times the base times the height. So, for example, if the base was equal to 5 and the height was equal to 6, then our area would be 1/2 times 5 times 6, which is 1/2 times 30-- which is equal to 15. Now what is less well-known is how to figure out the area of a triangle when you're only given the sides of the triangle. When you aren't given the height. So, for example, how do you figure out a triangle where I just give you the lengths of the sides. Let's say that's side a, side b, and side c. a, b, and c are the lengths of these sides. How do you figure that out? And to do that we're going to apply something called Heron's Formula. And I'm not going to prove it in this video. I'm going to prove it in a future video. And really to prove it you already probably have the tools necessary. It's really just the Pythagorean theorem and a lot of hairy algebra. But I'm just going to show you the formula now and how to apply it, and then you'll hopefully appreciate that it's pretty simple and pretty easy to remember. And it can be a nice trick to impress people with. So Heron's Formula says first figure out this third variable S, which is essentially the perimeter of this triangle divided by 2. a plus b plus c, divided by 2. Then once you figure out S, the area of your triangle-- of this triangle right there-- is going to be equal to the square root of S-- this variable S right here that you just calculated-- times S minus a, times S minus b, times S minus c. That's Heron's Formula right there. This combination. Let me square it off for you. So that right there is Heron's Formula. And if that looks a little bit daunting-- it is a little bit more daunting, clearly, than just 1/2 times base times height. Let's do it with an actual example or two, and actually see this is actually not so bad. So let's say I have a triangle. I'll leave the formula up there. So let's say I have a triangle that has sides of length 9, 11, and 16. So let's apply Heron's Formula. S in this situation is going to be the perimeter divided by 2. So 9 plus 11 plus 16, divided by 2. Which is equal to 9 plus 11-- is 20-- plus 16 is 36, divided by 2 is 18. And then the area by Heron's Formula is going to be equal to the square root of S-- 18-- times S minus a-- S minus 9. 18 minus 9, times 18 minus 11, times 18 minus 16. And then this is equal to the square root of 18 times 9 times 7 times 2. Which is equal to-- let's see, 2 times 18 is 36. So I'll just rearrange it a bit. This is equal to the square root of 36 times 9 times 7, which is equal to the square root of 36 times the square root of 9 times the square root of 7. The square root of 36 is just 6. This is just 3. And we don't deal with the negative square roots, because you can't have negative side lengths. And so this is going to be equal to 18 times the square root of 7. So just like that, you saw it, it only took a couple of minutes to apply Heron's Formula, or even less than that, to figure out that the area of this triangle right here is equal to 18 square root of seven. Anyway, hopefully you found that pretty neat.