[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.55,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.55,0:00:03.24,Default,,0000,0000,0000,,I think it's pretty common\Nknowledge how to find the area
Dialogue: 0,0:00:03.24,0:00:06.03,Default,,0000,0000,0000,,of the triangle if we know the\Nlength of its base
Dialogue: 0,0:00:06.03,0:00:07.25,Default,,0000,0000,0000,,and its height.
Dialogue: 0,0:00:07.25,0:00:10.54,Default,,0000,0000,0000,,So, for example, if that's my\Ntriangle, and this length right
Dialogue: 0,0:00:10.54,0:00:14.91,Default,,0000,0000,0000,,here-- this base-- is of length\Nb and the height right here is
Dialogue: 0,0:00:14.91,0:00:19.08,Default,,0000,0000,0000,,of length h, it's pretty common\Nknowledge that the area of this
Dialogue: 0,0:00:19.08,0:00:23.17,Default,,0000,0000,0000,,triangle is going to be equal\Nto 1/2 times the base
Dialogue: 0,0:00:23.17,0:00:24.44,Default,,0000,0000,0000,,times the height.
Dialogue: 0,0:00:24.44,0:00:30.24,Default,,0000,0000,0000,,So, for example, if the base\Nwas equal to 5 and the height
Dialogue: 0,0:00:30.24,0:00:37.18,Default,,0000,0000,0000,,was equal to 6, then our area\Nwould be 1/2 times 5 times 6,
Dialogue: 0,0:00:37.18,0:00:41.77,Default,,0000,0000,0000,,which is 1/2 times 30--\Nwhich is equal to 15.
Dialogue: 0,0:00:41.77,0:00:45.12,Default,,0000,0000,0000,,Now what is less well-known is\Nhow to figure out the area of a
Dialogue: 0,0:00:45.12,0:00:48.25,Default,,0000,0000,0000,,triangle when you're only given\Nthe sides of the triangle.
Dialogue: 0,0:00:48.25,0:00:49.74,Default,,0000,0000,0000,,When you aren't\Ngiven the height.
Dialogue: 0,0:00:49.74,0:00:53.47,Default,,0000,0000,0000,,So, for example, how do\Nyou figure out a triangle
Dialogue: 0,0:00:53.47,0:00:55.57,Default,,0000,0000,0000,,where I just give you the\Nlengths of the sides.
Dialogue: 0,0:00:55.57,0:01:00.53,Default,,0000,0000,0000,,Let's say that's side a, side\Nb, and side c. a, b, and c are
Dialogue: 0,0:01:00.53,0:01:01.64,Default,,0000,0000,0000,,the lengths of these sides.
Dialogue: 0,0:01:01.64,0:01:03.36,Default,,0000,0000,0000,,How do you figure that out?
Dialogue: 0,0:01:03.36,0:01:05.27,Default,,0000,0000,0000,,And to do that we're\Ngoing to apply something
Dialogue: 0,0:01:05.27,0:01:06.43,Default,,0000,0000,0000,,called Heron's Formula.
Dialogue: 0,0:01:06.43,0:01:12.21,Default,,0000,0000,0000,,
Dialogue: 0,0:01:12.21,0:01:13.79,Default,,0000,0000,0000,,And I'm not going to\Nprove it in this video.
Dialogue: 0,0:01:13.79,0:01:15.20,Default,,0000,0000,0000,,I'm going to prove it\Nin a future video.
Dialogue: 0,0:01:15.20,0:01:17.40,Default,,0000,0000,0000,,And really to prove it you\Nalready probably have
Dialogue: 0,0:01:17.40,0:01:18.72,Default,,0000,0000,0000,,the tools necessary.
Dialogue: 0,0:01:18.72,0:01:20.48,Default,,0000,0000,0000,,It's really just the\NPythagorean theorem and
Dialogue: 0,0:01:20.48,0:01:22.22,Default,,0000,0000,0000,,a lot of hairy algebra.
Dialogue: 0,0:01:22.22,0:01:24.23,Default,,0000,0000,0000,,But I'm just going to show you\Nthe formula now and how to
Dialogue: 0,0:01:24.23,0:01:26.76,Default,,0000,0000,0000,,apply it, and then you'll\Nhopefully appreciate that it's
Dialogue: 0,0:01:26.76,0:01:28.59,Default,,0000,0000,0000,,pretty simple and pretty\Neasy to remember.
Dialogue: 0,0:01:28.59,0:01:31.66,Default,,0000,0000,0000,,And it can be a nice trick\Nto impress people with.
Dialogue: 0,0:01:31.66,0:01:36.32,Default,,0000,0000,0000,,So Heron's Formula says first\Nfigure out this third variable
Dialogue: 0,0:01:36.32,0:01:38.64,Default,,0000,0000,0000,,S, which is essentially\Nthe perimeter of this
Dialogue: 0,0:01:38.64,0:01:40.66,Default,,0000,0000,0000,,triangle divided by 2.
Dialogue: 0,0:01:40.66,0:01:45.81,Default,,0000,0000,0000,,a plus b plus c, divided by 2.
Dialogue: 0,0:01:45.81,0:01:49.48,Default,,0000,0000,0000,,Then once you figure out S, the\Narea of your triangle-- of this
Dialogue: 0,0:01:49.48,0:01:55.84,Default,,0000,0000,0000,,triangle right there-- is going\Nto be equal to the square root
Dialogue: 0,0:01:55.84,0:01:59.71,Default,,0000,0000,0000,,of S-- this variable S right\Nhere that you just calculated--
Dialogue: 0,0:01:59.71,0:02:10.54,Default,,0000,0000,0000,,times S minus a, times S\Nminus b, times S minus c.
Dialogue: 0,0:02:10.54,0:02:12.48,Default,,0000,0000,0000,,That's Heron's\NFormula right there.
Dialogue: 0,0:02:12.48,0:02:13.83,Default,,0000,0000,0000,,This combination.
Dialogue: 0,0:02:13.83,0:02:16.13,Default,,0000,0000,0000,,Let me square it off for you.
Dialogue: 0,0:02:16.13,0:02:18.70,Default,,0000,0000,0000,,So that right there\Nis Heron's Formula.
Dialogue: 0,0:02:18.70,0:02:21.61,Default,,0000,0000,0000,,And if that looks a little bit\Ndaunting-- it is a little bit
Dialogue: 0,0:02:21.61,0:02:24.29,Default,,0000,0000,0000,,more daunting, clearly, than\Njust 1/2 times base
Dialogue: 0,0:02:24.29,0:02:25.29,Default,,0000,0000,0000,,times height.
Dialogue: 0,0:02:25.29,0:02:28.04,Default,,0000,0000,0000,,Let's do it with an actual\Nexample or two, and actually
Dialogue: 0,0:02:28.04,0:02:31.35,Default,,0000,0000,0000,,see this is actually\Nnot so bad.
Dialogue: 0,0:02:31.35,0:02:33.32,Default,,0000,0000,0000,,So let's say I have a triangle.
Dialogue: 0,0:02:33.32,0:02:35.30,Default,,0000,0000,0000,,I'll leave the\Nformula up there.
Dialogue: 0,0:02:35.30,0:02:37.46,Default,,0000,0000,0000,,So let's say I have a\Ntriangle that has sides
Dialogue: 0,0:02:37.46,0:02:44.92,Default,,0000,0000,0000,,of length 9, 11, and 16.
Dialogue: 0,0:02:44.92,0:02:47.04,Default,,0000,0000,0000,,So let's apply Heron's Formula.
Dialogue: 0,0:02:47.04,0:02:51.19,Default,,0000,0000,0000,,S in this situation is going to\Nbe the perimeter divided by 2.
Dialogue: 0,0:02:51.19,0:02:56.63,Default,,0000,0000,0000,,So 9 plus 11 plus\N16, divided by 2.
Dialogue: 0,0:02:56.63,0:03:00.43,Default,,0000,0000,0000,,Which is equal to 9 plus\N11-- is 20-- plus 16 is
Dialogue: 0,0:03:00.43,0:03:04.66,Default,,0000,0000,0000,,36, divided by 2 is 18.
Dialogue: 0,0:03:04.66,0:03:09.43,Default,,0000,0000,0000,,And then the area by Heron's\NFormula is going to be equal to
Dialogue: 0,0:03:09.43,0:03:19.38,Default,,0000,0000,0000,,the square root of S-- 18--\Ntimes S minus a-- S minus 9.
Dialogue: 0,0:03:19.38,0:03:27.79,Default,,0000,0000,0000,,18 minus 9, times 18 minus\N11, times 18 minus 16.
Dialogue: 0,0:03:27.79,0:03:31.49,Default,,0000,0000,0000,,
Dialogue: 0,0:03:31.49,0:03:38.20,Default,,0000,0000,0000,,And then this is equal to\Nthe square root of 18
Dialogue: 0,0:03:38.20,0:03:44.73,Default,,0000,0000,0000,,times 9 times 7 times 2.
Dialogue: 0,0:03:44.73,0:03:47.34,Default,,0000,0000,0000,,Which is equal to-- let's\Nsee, 2 times 18 is 36.
Dialogue: 0,0:03:47.34,0:03:48.90,Default,,0000,0000,0000,,So I'll just\Nrearrange it a bit.
Dialogue: 0,0:03:48.90,0:03:56.70,Default,,0000,0000,0000,,This is equal to the square\Nroot of 36 times 9 times 7,
Dialogue: 0,0:03:56.70,0:04:05.54,Default,,0000,0000,0000,,which is equal to the square\Nroot of 36 times the square
Dialogue: 0,0:04:05.54,0:04:09.33,Default,,0000,0000,0000,,root of 9 times the\Nsquare root of 7.
Dialogue: 0,0:04:09.33,0:04:14.13,Default,,0000,0000,0000,,The square root\Nof 36 is just 6.
Dialogue: 0,0:04:14.13,0:04:16.04,Default,,0000,0000,0000,,This is just 3.
Dialogue: 0,0:04:16.04,0:04:17.75,Default,,0000,0000,0000,,And we don't deal with the\Nnegative square roots,
Dialogue: 0,0:04:17.75,0:04:19.92,Default,,0000,0000,0000,,because you can't have\Nnegative side lengths.
Dialogue: 0,0:04:19.92,0:04:23.46,Default,,0000,0000,0000,,And so this is going to\Nbe equal to 18 times
Dialogue: 0,0:04:23.46,0:04:26.12,Default,,0000,0000,0000,,the square root of 7.
Dialogue: 0,0:04:26.12,0:04:28.06,Default,,0000,0000,0000,,So just like that, you saw it,\Nit only took a couple of
Dialogue: 0,0:04:28.06,0:04:30.76,Default,,0000,0000,0000,,minutes to apply Heron's\NFormula, or even less than
Dialogue: 0,0:04:30.76,0:04:33.42,Default,,0000,0000,0000,,that, to figure out that the\Narea of this triangle
Dialogue: 0,0:04:33.42,0:04:38.71,Default,,0000,0000,0000,,right here is equal to 18\Nsquare root of seven.
Dialogue: 0,0:04:38.71,0:04:42.04,Default,,0000,0000,0000,,Anyway, hopefully you\Nfound that pretty neat.
Dialogue: 0,0:04:42.04,0:04:42.33,Default,,0000,0000,0000,,