0:00:01.850,0:00:06.044
The theorem of Pythagoras is a[br]well known theorem.

0:00:06.600,0:00:11.192
It's also a very old one. Not[br]only does it bear the name of

0:00:11.192,0:00:14.800
Pythagoras in ancient Greek,[br]but it was also known to the

0:00:14.800,0:00:17.096
ancient Babylonians and to the[br]ancient Egyptians.

0:00:18.230,0:00:22.429
Most school boys school girls[br]learn of it as a square plus B

0:00:22.429,0:00:25.659
squared equals C squared, but[br]the actual statement of the

0:00:25.659,0:00:27.597
theorem is more to do with

0:00:27.597,0:00:31.988
areas. So let's have a look at[br]the statement of the theorem.

0:00:33.110,0:00:37.718
The theorem talks about[br]a right angle triangle.

0:00:39.270,0:00:43.222
So then we have a right angle[br]triangle and we show that with

0:00:43.222,0:00:46.870
this little mark here, so we[br]have a square in that corner.

0:00:47.700,0:00:52.661
What the theorem says is that[br]the square on the hypotenuse.

0:00:52.661,0:00:57.171
Now that means we have to[br]identify this side the

0:00:57.171,0:01:02.132
hypotenuse, and it's always the[br]longest side in the right angle

0:01:02.132,0:01:07.544
triangle and it's always the one[br]that is opposite to the right

0:01:07.544,0:01:12.505
angle. So it's this side here[br]which bears the name the

0:01:12.505,0:01:18.984
hypotenuse. So the theorem then[br]says the square on the

0:01:18.984,0:01:26.200
hypotenuse, so if they draw a[br]square on this hypotenuse, so

0:01:26.200,0:01:29.480
there we've got our square.

0:01:30.210,0:01:35.572
And it says that the area of[br]that Square is equal to the sum

0:01:35.572,0:01:36.721
of the areas.

0:01:37.300,0:01:42.684
Of the squares on the[br]two shorter sides.

0:01:43.820,0:01:47.130
So these are all squares.

0:01:47.130,0:01:52.539
All the same, all got right[br]angles in them.

0:01:53.120,0:01:55.766
This is a square as well.

0:01:56.290,0:02:02.252
All sides are equal. Got a[br]right angle. What this says

0:02:02.252,0:02:09.840
is that this area A plus this[br]area B is equal to that area.

0:02:09.840,0:02:12.008
C A+B equals C.

0:02:13.140,0:02:20.211
OK. Let's have a look at[br]a special case and see if we can

0:02:20.211,0:02:22.246
see how that actually works.

0:02:22.260,0:02:30.108
Got here a[br]single red triangle.

0:02:33.460,0:02:35.830
It's a very special triangle.

0:02:37.340,0:02:41.564
This side along the bottom is[br]one unit long and that side

0:02:41.564,0:02:46.492
along there is 2 units long, so[br]I'm just going to fix it. Little

0:02:46.492,0:02:47.900
bit of Blu tack.

0:02:48.780,0:02:54.510
On there I'm just going to fix[br]it so that in fact I can draw

0:02:54.510,0:03:01.696
round it. So that[br]we've drawn around that

0:03:01.696,0:03:09.584
triangle, now I want[br]to fix some squares

0:03:09.584,0:03:13.528
on the sides of

0:03:13.528,0:03:20.611
this triangle. So there's[br]the square on that side, but the

0:03:20.611,0:03:27.317
square on this side is going to[br]look in fact just a tiny bit

0:03:27.317,0:03:32.586
different. It's going to be an[br]assembly of triangles, each one

0:03:32.586,0:03:37.376
identical to this original[br]triangle, so there's half of it.

0:03:37.970,0:03:40.322
Now I'm going to put the other

0:03:40.322,0:03:43.370
half on. Here.

0:03:44.110,0:03:48.050
Lastly, debate.

0:03:49.380,0:03:53.660
I will make it up with[br]this bit.

0:03:56.260,0:04:00.676
OK, now remember what the[br]theorem says. It says that if we

0:04:00.676,0:04:05.828
take the area of this square and[br]add it to the area of that

0:04:05.828,0:04:09.876
square, we should get the area[br]of this square on the

0:04:09.876,0:04:11.716
hypotenuse. Let's see how right

0:04:11.716,0:04:17.641
that is. Each one of these red[br]triangles that I've got here and

0:04:17.641,0:04:21.832
that I'm putting on is exactly[br]the same as these triangles

0:04:21.832,0:04:24.880
here, so I can put one on there.

0:04:28.000,0:04:33.964
That one[br]on there.

0:04:33.964,0:04:36.946
Now put

0:04:36.946,0:04:39.928
the square.

0:04:40.240,0:04:45.085
In there.[br]That

0:04:45.085,0:04:50.515
one[br]in

0:04:50.515,0:04:57.312
there.[br]And then finally put

0:04:57.312,0:05:00.564
this one in there.

0:05:01.190,0:05:06.390
What we can see there is that[br]quite clearly the area of this

0:05:06.390,0:05:11.990
square is made up of the area of[br]that square plus the area of

0:05:11.990,0:05:17.240
that square. OK, if this theorem[br]is true, and certainly it seems

0:05:17.240,0:05:22.112
to be true for this, and it has[br]been proved that it's true, then

0:05:22.112,0:05:24.200
how do we get the traditional?

0:05:24.970,0:05:30.960
Schoolboy schoolgirl theorem. A[br]squared plus B squared equals C

0:05:30.960,0:05:37.926
squared. So draw our[br]right angle triangle.

0:05:39.760,0:05:43.376
Will put on the

0:05:43.376,0:05:46.239
squares. On the sides.

0:05:46.810,0:05:50.992
And[br]we

0:05:50.992,0:05:55.174
label[br]these

0:05:55.174,0:05:57.265
squares

0:05:57.265,0:06:03.370
A&B&C. We label[br]the lengths of the sides that

0:06:03.370,0:06:05.710
lay. It'll be a little C.

0:06:06.430,0:06:11.383
Then the theorem tells us that[br]if we add together the areas of

0:06:11.383,0:06:16.336
the squares on the two shorter[br]sides, the result is the area of

0:06:16.336,0:06:20.527
the square on the longest side[br]of the triangle the hypotenuse.

0:06:20.527,0:06:25.480
So A plus B is equal to see, but[br]this is a square.

0:06:26.200,0:06:32.248
And so it's area is A squared[br]and this is also a square, so

0:06:32.248,0:06:34.408
its area is B squared.

0:06:34.940,0:06:41.870
This is a square and its area is[br]C squared and so we have the

0:06:41.870,0:06:45.525
traditional result. Always when[br]answering questions on

0:06:45.525,0:06:49.278
Pythagoras Theorem, make sure[br]that you read the question.

0:06:49.278,0:06:53.448
Don't just write down this[br]result Willy nilly. Make sure

0:06:53.448,0:06:56.784
you've actually read the[br]question and that you're

0:06:56.784,0:07:01.788
thinking about which sides are[br]which. Let's have a look at one

0:07:01.788,0:07:03.039
or two examples.

0:07:03.580,0:07:10.330
First of all, let's take[br]a triangle whose two shorter

0:07:10.330,0:07:16.405
sides are of length three[br]and of length 4.

0:07:16.920,0:07:22.668
And the question is, how long is[br]the longest side the hypotenuse?

0:07:23.330,0:07:27.620
So we can label our sides, let's[br]call this one a.

0:07:28.220,0:07:33.694
Call this one B and then the one[br]we want to find is see.

0:07:34.520,0:07:42.308
So using the theorem, we know[br]that A squared plus B squared.

0:07:42.310,0:07:43.810
Is C squared?

0:07:44.520,0:07:51.630
A is 3 three squared[br]plus B is 4, four

0:07:51.630,0:07:54.474
squared equals C squared.

0:07:55.300,0:08:01.660
3 squared is 9 and 4 squared. Is[br]16 equals C squared, and so 25

0:08:01.660,0:08:08.020
is equal to C squared, and so we[br]now need to take the square root

0:08:08.020,0:08:12.260
of both sides, and so five is[br]equal to see.

0:08:13.160,0:08:18.800
That's the length of the[br]hypotenuse. The longest side of

0:08:18.800,0:08:24.962
the triangle. Let's take another[br]example. This time one where we

0:08:24.962,0:08:30.938
know the hypotenuse. Let's say[br]it's of length 13 and we know

0:08:30.938,0:08:37.412
one of the shorter sides. Let's[br]say that's of length 5, but we

0:08:37.412,0:08:40.898
want to know is what's the other

0:08:40.898,0:08:48.067
shorter side. Let's label these.[br]If I call this One X, let's say,

0:08:48.067,0:08:51.476
and this one, why I'm call this

0:08:51.476,0:08:56.942
one zed. Because I've label them[br]differently, doesn't make the

0:08:56.942,0:09:01.632
relationship any different. It[br]still X squared, plus Y squared

0:09:01.632,0:09:07.260
equals zed squared. The sum of[br]the squares on the two shorter

0:09:07.260,0:09:11.950
sides is equal to the square on[br]the hypotenuse. Let's

0:09:11.950,0:09:17.578
substituting. Now X is 5, so[br]that's 5 squared plus Y squared.

0:09:17.578,0:09:23.206
That's what we're trying to find[br]is equal to zed squared and

0:09:23.206,0:09:26.210
said. Is 13, so that's 13

0:09:26.210,0:09:32.515
squared. So 25[br]plus Y squared

0:09:32.515,0:09:35.650
is equal to

0:09:35.650,0:09:42.388
169. And so[br]Y squared must be

0:09:42.388,0:09:44.797
equal to 144.

0:09:45.570,0:09:51.464
And now if we just turn over the[br]page and take that last line

0:09:51.464,0:09:56.937
again, Y squared is equal to[br]144. We need to take the square

0:09:56.937,0:10:02.410
root of both sides. So why must[br]be the square root of 144?

0:10:02.510,0:10:06.180
12 Will do.

0:10:06.700,0:10:13.382
One more. Let's take our[br]right angle triangle here, like

0:10:13.382,0:10:20.411
this. And let's say this is[br]17 and this is 15.

0:10:20.930,0:10:27.762
This one is the one we want to[br]know now. It's still a right

0:10:27.762,0:10:32.642
angle triangle, the hypotenuse[br]is still the side that is

0:10:32.642,0:10:38.010
opposite to the right angle.[br]Still, the longest side in the

0:10:38.010,0:10:40.450
triangle. Let's label our sides.

0:10:41.340,0:10:46.644
And again the labelings[br]different but the relationship

0:10:46.644,0:10:54.600
remains the same P squared plus[br]Q Squared is equal to R-squared.

0:10:54.600,0:10:57.915
So let's substitute in our

0:10:57.915,0:11:01.370
numbers. P squared that's

0:11:01.370,0:11:08.315
15 squared. Q is what[br]we're trying to find sales plus

0:11:08.315,0:11:15.140
Q squared is equal to R-squared[br]and R is 17. So we have

0:11:15.140,0:11:18.290
17 squared. Now we need some

0:11:18.290,0:11:21.196
numbers. Q

0:11:21.196,0:11:29.111
squared equals.[br]15 squared, that's

0:11:29.111,0:11:36.180
225. And 17 squared[br]is 289. You can work these two

0:11:36.180,0:11:41.570
out on a Calculator or you can[br]use long multiplication. Doesn't

0:11:41.570,0:11:44.510
matter which. Now let's find Q

0:11:44.510,0:11:50.650
squared. That's equal to 299[br]takeaway 225, which is 64. And

0:11:50.650,0:11:56.249
so taking the square roots again[br]Q is equal to 8.

0:11:57.330,0:12:03.028
In each of these three, the[br]answers of being exact, they've

0:12:03.028,0:12:05.100
been whole number values.

0:12:05.930,0:12:11.000
And these whole number triples[br]or pythagorean triples as we

0:12:11.000,0:12:13.910
call them. Occur quite

0:12:13.910,0:12:18.091
frequently. But you probably[br]won't be lucky when you do

0:12:18.091,0:12:21.142
questions you in getting an[br]exact answer, you almost

0:12:21.142,0:12:24.871
certainly have to use a[br]Calculator for many of them, and

0:12:24.871,0:12:29.278
you have to decide to shorten[br]the answer to a given number of

0:12:29.278,0:12:32.668
decimal places or a given number[br]of significant figures. I've

0:12:32.668,0:12:36.736
used whole numbers 'cause it's[br]easier for me to work with them

0:12:36.736,0:12:40.804
on this page, but do remember[br]they won't always be exact. In

0:12:40.804,0:12:44.872
fact, more than likely they[br]won't be exact and you will have

0:12:44.872,0:12:46.228
to work them out.

0:12:46.270,0:12:47.629
Using a Calculator.

0:12:48.350,0:12:52.679
Let's have a look at another[br]application of Pythagoras.

0:12:53.220,0:12:59.720
Supposing we know the dimensions[br]of a box, let's say we know that

0:12:59.720,0:13:01.720
it's 3 by 4.

0:13:02.390,0:13:05.450
By 12 so that means[br]we know that it's.

0:13:06.690,0:13:07.690
Three wide.

0:13:08.840,0:13:12.260
For long. And.

0:13:13.540,0:13:14.500
12 high

0:13:15.660,0:13:18.660
Let's complete our box.

0:13:19.640,0:13:27.080
So it's 3 by 4[br]by 12 high. Let's make

0:13:27.080,0:13:34.520
it look a little bit[br]3 dimensional by putting in.

0:13:35.340,0:13:38.430
In the dotted line, the bits[br]that we can't see.

0:13:39.530,0:13:44.808
Question. How big is the[br]diagonal of the box? The

0:13:44.808,0:13:49.632
diagonal not that runs across a[br]face but runs from one corner

0:13:49.632,0:13:53.652
across to another corner. How[br]can we work this out?

0:13:54.510,0:14:01.179
Which is a cuboid. Each of these[br]sides is a rectangle. All the

0:14:01.179,0:14:06.309
joints are at right angles.[br]Let's put this diagonal in.

0:14:07.190,0:14:12.974
So there it goes spanning the[br]box well across here, down to

0:14:12.974,0:14:17.794
here and across here we've got[br]another right angle triangle.

0:14:18.960,0:14:24.308
So we know that if we knew that,[br]we could find that. But this

0:14:24.308,0:14:29.274
gives us a right angle triangle[br]down here. And we do know these

0:14:29.274,0:14:33.858
two sides, so let's see if we[br]can work that out. Let's.

0:14:34.400,0:14:38.998
Make clear we've got a right[br]angle triangle here and one

0:14:38.998,0:14:46.876
here. So first of all, how[br]long is that? How long is X

0:14:46.876,0:14:51.898
the diagonal of this base[br]rectangle here? Well, pythagoras

0:14:51.898,0:14:58.594
tells us that 3 squared +4[br]squared is equal to X squared,

0:14:58.594,0:15:01.384
so that's 9 + 16.

0:15:02.130,0:15:08.550
Is 25 and that must mean that[br]X is equal to 5.

0:15:09.290,0:15:15.744
So we've now got X is 5 and[br]we can use Pythagoras again in

0:15:15.744,0:15:22.786
this triangle. 5 squared plus[br]12 squared is equal to the

0:15:22.786,0:15:29.100
length of the diagonal squared.[br]Let's call that Y equals Y

0:15:29.100,0:15:35.988
squared, so we have 25 +[br]144 is equal to Y squared,

0:15:35.988,0:15:42.876
so that's 169 is equal to[br]Y squared. So why must be

0:15:42.876,0:15:46.320
the square root of 169, which

0:15:46.320,0:15:51.790
is 13? So we can use the[br]theorem of Pythagoras in three

0:15:51.790,0:15:55.966
dimensions. We can use it to[br]solve problems that are set up

0:15:55.966,0:15:57.358
in 3 dimensional objects.

0:15:58.060,0:16:01.518
What about making use of it in

0:16:01.518,0:16:04.980
other ways? What about

0:16:04.980,0:16:10.968
cartesian geometry? Cartesian[br]geometry is geometry that set

0:16:10.968,0:16:17.298
out on a plane that's got[br]Cartesian coordinates 11 the

0:16:17.298,0:16:19.830
.22, and so on.

0:16:20.510,0:16:24.740
One of the other things about[br]Pythagoras theorem is this.

0:16:25.610,0:16:32.360
It tells us if we[br]have a particular triangle, it

0:16:32.360,0:16:37.085
tells us the relationship[br]between the sides.

0:16:38.020,0:16:41.626
And it works in reverse if.

0:16:42.520,0:16:44.020
It's.

0:16:44.950,0:16:52.722
The area.[br]Of the

0:16:52.722,0:16:59.956
square. On[br]the longest

0:16:59.956,0:17:03.730
side. Is

0:17:03.730,0:17:05.540
equal.

0:17:06.050,0:17:11.724
Tool.[br]Area of

0:17:11.724,0:17:14.628
the squares.

0:17:15.870,0:17:19.542
On the two

0:17:19.542,0:17:24.433
shorter sides.[br]Of

0:17:24.433,0:17:26.876
a

0:17:26.876,0:17:29.319
triangle.

0:17:29.320,0:17:36.362
Then[br]The triangle

0:17:36.362,0:17:39.854
is right

0:17:39.854,0:17:45.974
angled. So this is turned[br]the theorem round. Instead of

0:17:45.974,0:17:51.100
saying if the triangles right[br]angled, then this is So what

0:17:51.100,0:17:56.692
it's saying is if this is so,[br]then the triangle is right

0:17:56.692,0:18:02.284
angled and we can make use of[br]this in areas like Cartesian

0:18:02.284,0:18:07.410
coordinate geometry to find out[br]whether or not a triangle is

0:18:07.410,0:18:09.740
right angled or not so.

0:18:10.260,0:18:14.004
Let's take the three

0:18:14.004,0:18:16.760
points. 34

0:18:17.390,0:18:23.425
26[br]One note and let's see

0:18:23.425,0:18:26.710
if they do give us a[br]right angle triangle.

0:18:28.230,0:18:33.474
So first of all, let's plot[br]these points. First of all, will

0:18:33.474,0:18:35.659
mark off on the axes.

0:18:36.890,0:18:39.359
Roughly equal spaces.

0:18:40.890,0:18:45.270
The[br]same

0:18:45.270,0:18:47.460
markings.

0:18:48.660,0:18:54.740
On the Y axis, again[br]roughly equal spaces.

0:18:55.270,0:18:59.768
And now we're plot the points.[br]The point one note will be here.

0:18:59.768,0:19:01.844
Will call that the point a.

0:19:02.800,0:19:08.680
The Point B will be the .26 that[br]will be roughly there.

0:19:09.270,0:19:16.230
And the Point C will be the .3[br]four and so that will be three

0:19:16.230,0:19:19.014
and four that will be roughly

0:19:19.014,0:19:21.878
there. Let's join up.

0:19:22.490,0:19:23.490
These points.

0:19:25.790,0:19:28.850
To give us a triangle.

0:19:29.730,0:19:33.450
Now it doesn't look right angle,[br]but we don't really know 'cause.

0:19:33.450,0:19:35.000
These were only rough markings.

0:19:35.790,0:19:40.782
Is it a right angle triangle or[br]not? You mustn't believe what

0:19:40.782,0:19:45.358
our eyes tell us, especially[br]when we can do calculation which

0:19:45.358,0:19:47.854
will help us to be exact.

0:19:47.870,0:19:53.148
So I need to work out the[br]lengths of the sides of the

0:19:53.148,0:19:55.178
triangles, so let me surround.

0:19:57.900,0:19:59.829
Is triangle with.

0:20:00.330,0:20:01.890
A rectangle.

0:20:06.240,0:20:10.490
And because I've surrounded by a[br]rectangle, these are all.

0:20:11.050,0:20:12.460
Right angles here.

0:20:13.140,0:20:16.716
And I can see now what the[br]lengths of these little bits

0:20:16.716,0:20:22.431
are. So let me workout AB[br]squared to begin with. a B

0:20:22.431,0:20:27.723
squared. Now I can do that[br]because this side here is of

0:20:27.723,0:20:32.574
length 6, so that's 6 squared[br]plus this length squared here,

0:20:32.574,0:20:34.779
which is a blank one.

0:20:35.700,0:20:40.500
That's 36 plus one is 37, so[br]I've used Pythagoras in this

0:20:40.500,0:20:46.500
right angle triangle to find out[br]what a B is, or at least to find

0:20:46.500,0:20:51.700
out what AB squared is. Let me[br]do that again here, this time

0:20:51.700,0:20:56.900
for the side BCBC squared is and[br]I'm going to use this right

0:20:56.900,0:21:02.100
angle triangle here, and again I[br]can see that this length here is

0:21:02.100,0:21:06.500
one, so that's one squared plus[br]the square of this length.

0:21:06.540,0:21:08.708
So that's 2 squared.

0:21:09.300,0:21:16.110
That's 1 + 4 gives[br]main 5. Finally, let's do

0:21:16.110,0:21:18.153
a C squared.

0:21:19.230,0:21:25.796
And again, I can see that this[br]length here is of length 4, so

0:21:25.796,0:21:31.424
that's 4 squared plus the square[br]of this length here. That's 2

0:21:31.424,0:21:37.052
squared is equal to 16 + 4, and[br]that gives me 20.

0:21:37.580,0:21:43.612
Now, if this is a right angle[br]triangle, if I take the squares

0:21:43.612,0:21:48.716
of the lengths of the two[br]shorter sides, that's a CNBC.

0:21:50.030,0:21:54.320
Add the squares of those[br]lens together. I should get

0:21:54.320,0:21:59.039
the square of the length of[br]the third side. If it's

0:21:59.039,0:22:03.758
right angle and if I don't[br]then it's not right angle,

0:22:03.758,0:22:09.335
so let's do that AC squared[br]plus BC squared is 5 + 20.

0:22:11.150,0:22:17.520
5 for BC squared, 20 for AC[br]square, and that gives me 25.

0:22:19.130,0:22:26.730
But a B squared is[br]37. That does not equal

0:22:26.730,0:22:34.330
37 and therefore the triangle[br]ABC is not right angle.

0:22:34.340,0:22:41.534
The fact that Pythagoras theorem[br]is about squares is fairly well

0:22:41.534,0:22:48.074
known. What's not as well known,[br]although it's fairly obvious

0:22:48.074,0:22:50.690
once you've seen it.

0:22:51.360,0:22:57.864
Is that if you take any regular[br]figure or similar figures and

0:22:57.864,0:23:03.284
place them on the sides of a[br]right angle triangle?

0:23:03.970,0:23:10.390
Then the area of the figure[br]on the hypotenuse is equal to

0:23:10.390,0:23:17.345
the sum of the areas of the[br]same figures on the other two

0:23:17.345,0:23:18.415
shorter sides.

0:23:19.620,0:23:24.930
Let's make it perhaps as[br]dramatic as we can. Supposing

0:23:24.930,0:23:30.771
report semi circles on the sides[br]of this right angle triangle.

0:23:31.610,0:23:38.666
Is it the case that the area[br]of the semi circles on the two

0:23:38.666,0:23:44.714
shorter sides add up to the area[br]of the semicircle on the

0:23:44.714,0:23:49.250
hypotenuse? Is that so? Well,[br]let's label the sides.

0:23:49.890,0:23:53.306
Play, it'll be, it'll

0:23:53.306,0:24:00.766
see. And workout the area area[br]of a is now the area of a

0:24:00.766,0:24:06.642
semicircle is half the area of[br]the circle and the area of a

0:24:06.642,0:24:12.066
circle is pie are squared π[br]times the square of the radius.

0:24:12.066,0:24:18.394
Now the diameter of the circle[br]is a, so the radius is a over

0:24:18.394,0:24:20.202
2, so that's π.

0:24:21.040,0:24:27.424
Times a over 2 squared. If we[br]work that out, that's Pi over 8A

0:24:27.424,0:24:33.352
squared, over 2 all squared is a[br]squared over 4. That's where the

0:24:33.352,0:24:39.736
a square comes from and the two[br]times by the four gives us the

0:24:39.736,0:24:46.416
8. Be what's the area[br]of this semicircle here? Well,

0:24:46.416,0:24:52.700
again. The diameter of the[br]semicircle is B, so the radius

0:24:52.700,0:24:59.896
is be over 2, so the area[br]is 1/2 Pi R-squared and are. We

0:24:59.896,0:25:07.092
just said was be over 2, so[br]we square that and we have π

0:25:07.092,0:25:09.662
over 8B squared, this one.

0:25:10.590,0:25:18.160
Same again, C equals 1/2.[br]Thai R-squared and R is

0:25:18.160,0:25:25.730
1/2 the diameter 1/2 of[br]CC over 2 all squared,

0:25:25.730,0:25:29.515
and so that's Pi over

0:25:29.515,0:25:36.845
8C squared. So let's add[br]these together A&B and see what

0:25:36.845,0:25:44.615
we get a is π over 8A[br]squared. B is π over 8B squared,

0:25:44.615,0:25:51.830
and there's a common factor here[br]of π over 8. We can take

0:25:51.830,0:25:57.380
out, leaving us with A squared[br]plus B squared, but.

0:25:58.000,0:26:02.059
The theorem of Pythagoras tells[br]us that in a right angle

0:26:02.059,0:26:06.118
triangle, a squared plus B[br]squared is C squared, and so

0:26:06.118,0:26:08.701
that's π over 8 times by C

0:26:08.701,0:26:11.900
squared. Which is just say.

0:26:12.580,0:26:18.417
And so the question that we[br]asked at the top is true. This

0:26:18.417,0:26:22.907
does work for semi circles. It[br]would work for equilateral

0:26:22.907,0:26:27.397
triangles. It would work for[br]hexagons. It would work for

0:26:27.397,0:26:29.193
similar isosceles triangles and

0:26:29.193,0:26:34.714
so on. Pythagoras theorem is[br]quite a wonderful theorem, and

0:26:34.714,0:26:39.644
there's quite a lot to be gained[br]from exploring pythagorean

0:26:39.644,0:26:46.053
triple, such as three 455-1213[br]and the other one that we saw in

0:26:46.053,0:26:48.025
this lecture, eight 1570.