Geometry (all content)
Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides. Created by Sal Khan.
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- Can you use this formula for other shapes or just triangles?(102 votes)
- Technically you can divide any polygon (not circles or anything that's built from non-straight lines) to triangles. For instance, a square is just two (equal) triangles together. A pentagon forms three triangles - and so on. This is why the triangle is one of the most basic shapes in geometry - simply because it can form everything.(121 votes)
- why did you divide the perimeter by 2? isnt a+b+c already the perimeter?(44 votes)
- when we have to find the "s" , we have to divide the perimeter by 2(5 votes)
- At1:38, why's perimeter represented with an S and not a P?(13 votes)
- it's just a variable... variables can change... and plus, it's meaning is semi-perimeter, which means half-of-the-triangle...i.e., a+b+c / 2(1 vote)
- Why is it called heron in the first place and when was it created?(5 votes)
- It's called Heron's Formula because it is credited to the Heron (nowadays Hero) of Alexandria and proof is found in his book called Metrica written A.D. 60. A hypothesis says that Archemedes (Greek warrior) knew it over 200 years earlier and that is possible.(6 votes)
- does this formula work for all types of triangles(5 votes)
- Yes, the formula works for every type of triangle(3 votes)
- how to find the sides when you have the area given. if the area is 4 root 5(4 votes)
- Two triangles with different side measures could have different areas. Therefore, the side measures of a triangle can not be found given only the area.(4 votes)
- at1:29what does Sal mean by "nice trick"?(2 votes)
- He's referring to the formula (Heron's formula) itself.(4 votes)
- Why do we need to find S in the equation..
I know that S=a+b+c/2
Couldn't we just do that in the old way like b*h/2
Why should we use this long and difficult equation?(2 votes)
- The reason we need to know what S is is because we do not know the height. This formula is actually a really easy way to solve find the area of a triangle once you know you to use it. Another was would be to divide a triangle into two right triangles. You could then use the Law of Cosines (https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/law-sines-cosines/v/law-of-cosines-example) to find the angle measures. You could then solve for the height, which would be the dividing line through the middle of the triangle, by using the trigonometric functions (sin, cos, tan, etc.). After doing all this, you could solve for the area of the triangle using A=b*h/2. However, Heron's Formula is actually much shorter and less difficult than solving it the other way. If you put some effort into learning this formula, maybe even watch Sal's video on proofing the formula, then you will find this to be a very helpful and easy to use formula.(4 votes)
- Can you do it with lengths of 10, 6 and 4? Because I seem to end up with an area of 0. Am I doing something wrong or is the formula flawed?(1 vote)
- Neither. A "triangle" with side lengths of 10, 6, and 4 would be perfectly flat and have no area. (It isn't really a triangle at all.) For three side lengths to form a triangle, the sum of the two smaller sides must be LARGER than the largest side.(5 votes)
- why can't you multiply 18*9*7* 2?(3 votes)
- I'm just going to reroute you to Johan Kanselaar's answer.
He could have. It would have been the same answer. But he's looking for intermediate numbers which he can squareroot easily. 18x2 = 36. sqrt(36) = 6. sqrt(9) = 3.
Because of the commutative property of multiplication (A*B = B*A or in this case a*b*c*d = d*a*b*c) he can swap any of the four numbers being multiplied around.
He can also do 18*9*7*2 = 18*18*7 because 9*2 = 18. sqrt(18*18*7) is like sqrt(18*18) * sqrt(7) which is 18*sqrt(7). 18*sqrt(7) = sqrt(2268)(1 vote)
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.