0:00:00.000,0:00:00.550


0:00:00.550,0:00:03.240
I think it's pretty common[br]knowledge how to find the area

0:00:03.240,0:00:06.030
of the triangle if we know the[br]length of its base

0:00:06.030,0:00:07.250
and its height.

0:00:07.250,0:00:10.540
So, for example, if that's my[br]triangle, and this length right

0:00:10.540,0:00:14.910
here-- this base-- is of length[br]b and the height right here is

0:00:14.910,0:00:19.080
of length h, it's pretty common[br]knowledge that the area of this

0:00:19.080,0:00:23.170
triangle is going to be equal[br]to 1/2 times the base

0:00:23.170,0:00:24.440
times the height.

0:00:24.440,0:00:30.240
So, for example, if the base[br]was equal to 5 and the height

0:00:30.240,0:00:37.180
was equal to 6, then our area[br]would be 1/2 times 5 times 6,

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which is 1/2 times 30--[br]which is equal to 15.

0:00:41.770,0:00:45.120
Now what is less well-known is[br]how to figure out the area of a

0:00:45.120,0:00:48.250
triangle when you're only given[br]the sides of the triangle.

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When you aren't[br]given the height.

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So, for example, how do[br]you figure out a triangle

0:00:53.470,0:00:55.570
where I just give you the[br]lengths of the sides.

0:00:55.570,0:01:00.530
Let's say that's side a, side[br]b, and side c. a, b, and c are

0:01:00.530,0:01:01.640
the lengths of these sides.

0:01:01.640,0:01:03.360
How do you figure that out?

0:01:03.360,0:01:05.270
And to do that we're[br]going to apply something

0:01:05.270,0:01:06.430
called Heron's Formula.

0:01:06.430,0:01:12.210


0:01:12.210,0:01:13.790
And I'm not going to[br]prove it in this video.

0:01:13.790,0:01:15.200
I'm going to prove it[br]in a future video.

0:01:15.200,0:01:17.400
And really to prove it you[br]already probably have

0:01:17.400,0:01:18.720
the tools necessary.

0:01:18.720,0:01:20.480
It's really just the[br]Pythagorean theorem and

0:01:20.480,0:01:22.220
a lot of hairy algebra.

0:01:22.220,0:01:24.230
But I'm just going to show you[br]the formula now and how to

0:01:24.230,0:01:26.760
apply it, and then you'll[br]hopefully appreciate that it's

0:01:26.760,0:01:28.590
pretty simple and pretty[br]easy to remember.

0:01:28.590,0:01:31.660
And it can be a nice trick[br]to impress people with.

0:01:31.660,0:01:36.320
So Heron's Formula says first[br]figure out this third variable

0:01:36.320,0:01:38.640
S, which is essentially[br]the perimeter of this

0:01:38.640,0:01:40.660
triangle divided by 2.

0:01:40.660,0:01:45.810
a plus b plus c, divided by 2.

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Then once you figure out S, the[br]area of your triangle-- of this

0:01:49.480,0:01:55.840
triangle right there-- is going[br]to be equal to the square root

0:01:55.840,0:01:59.710
of S-- this variable S right[br]here that you just calculated--

0:01:59.710,0:02:10.540
times S minus a, times S[br]minus b, times S minus c.

0:02:10.540,0:02:12.480
That's Heron's[br]Formula right there.

0:02:12.480,0:02:13.830
This combination.

0:02:13.830,0:02:16.130
Let me square it off for you.

0:02:16.130,0:02:18.700
So that right there[br]is Heron's Formula.

0:02:18.700,0:02:21.610
And if that looks a little bit[br]daunting-- it is a little bit

0:02:21.610,0:02:24.290
more daunting, clearly, than[br]just 1/2 times base

0:02:24.290,0:02:25.290
times height.

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Let's do it with an actual[br]example or two, and actually

0:02:28.040,0:02:31.350
see this is actually[br]not so bad.

0:02:31.350,0:02:33.320
So let's say I have a triangle.

0:02:33.320,0:02:35.300
I'll leave the[br]formula up there.

0:02:35.300,0:02:37.460
So let's say I have a[br]triangle that has sides

0:02:37.460,0:02:44.920
of length 9, 11, and 16.

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So let's apply Heron's Formula.

0:02:47.040,0:02:51.190
S in this situation is going to[br]be the perimeter divided by 2.

0:02:51.190,0:02:56.630
So 9 plus 11 plus[br]16, divided by 2.

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Which is equal to 9 plus[br]11-- is 20-- plus 16 is

0:03:00.430,0:03:04.660
36, divided by 2 is 18.

0:03:04.660,0:03:09.430
And then the area by Heron's[br]Formula is going to be equal to

0:03:09.430,0:03:19.380
the square root of S-- 18--[br]times S minus a-- S minus 9.

0:03:19.380,0:03:27.790
18 minus 9, times 18 minus[br]11, times 18 minus 16.

0:03:27.790,0:03:31.490


0:03:31.490,0:03:38.200
And then this is equal to[br]the square root of 18

0:03:38.200,0:03:44.730
times 9 times 7 times 2.

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Which is equal to-- let's[br]see, 2 times 18 is 36.

0:03:47.340,0:03:48.900
So I'll just[br]rearrange it a bit.

0:03:48.900,0:03:56.700
This is equal to the square[br]root of 36 times 9 times 7,

0:03:56.700,0:04:05.540
which is equal to the square[br]root of 36 times the square

0:04:05.540,0:04:09.330
root of 9 times the[br]square root of 7.

0:04:09.330,0:04:14.130
The square root[br]of 36 is just 6.

0:04:14.130,0:04:16.040
This is just 3.

0:04:16.040,0:04:17.750
And we don't deal with the[br]negative square roots,

0:04:17.750,0:04:19.920
because you can't have[br]negative side lengths.

0:04:19.920,0:04:23.460
And so this is going to[br]be equal to 18 times

0:04:23.460,0:04:26.120
the square root of 7.

0:04:26.120,0:04:28.060
So just like that, you saw it,[br]it only took a couple of

0:04:28.060,0:04:30.760
minutes to apply Heron's[br]Formula, or even less than

0:04:30.760,0:04:33.420
that, to figure out that the[br]area of this triangle

0:04:33.420,0:04:38.710
right here is equal to 18[br]square root of seven.

0:04:38.710,0:04:42.040
Anyway, hopefully you[br]found that pretty neat.

0:04:42.040,0:04:42.331