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# Heron's formula

## Video transcript

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 if this 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 size 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 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 herons 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 too to impress people with so Heron's formula says first figure out 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 your area is going to be equal to the square root of s this variable s right here you just calculated times s minus a times s minus B times s minus C s minus C that's Heron's formula right there this komban let me square it off for you so that right there is herons format 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 that it's 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 the area by Heron's formula is going to be equal to the square root the square root of s 18 times s minus a s minus 9 18 minus 9 18 minus 9 times 18 minus 11 times 18 minus 16 18 minus 16 and then this becomes 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 which is equal to this 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 this is just 6 this is just 3 I mean you don't deal with the negative square roots because you can 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 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 the area is equal to 18 square roots of 7 anyway hopefully found that pretty neat