### Video Transcript

๐๐ด๐ต is a sector of a circle with
radius 12 centimetres. ๐ด๐ต is a chord of the sector. Calculate the area of the shaded
section, giving your answer accurate to three significant figures.

Now, the way that weโre actually
gonna solve this problem, Iโve kind of shown using these little diagrams because
what weโre gonna do is weโre gonna find the area of the total sector minus the area
of the triangle ๐ด๐๐ต. And this will equal our shaded
region. So what weโre gonna need to do
first of all is find out the area of the total sector and also find out the area of
the triangle.

So weโre gonna start by finding the
area of the whole sector. Well, we actually have the formula
to help us and that says that the area is equal to ๐ over 360 multiplied by ๐๐
squared. And thatโs where ๐ is the angle
thatโs within the sector. So what weโre saying is itโs a
fraction of our circle multiplied by ๐๐ squared. Okay, so now what weโre gonna do is
actually substitute our values in. And when we do that, we get ๐ด is
equal to 45 over 360. Thatโs because our angle is 45
degrees multiplied by ๐ multiplied by ๐ squared, so 12 squared.

Okay, great, now we can actually
calculate this. So when we actually do calculate
this, what weโre gonna get is that the area is equal to 144 over eight ๐. And we get that because we get 144
because 12 squared is 144 and then 45 over 360 gives us an eighth. So weโre left with 144 over eight
๐. So now, we can actually simplify
this. And when we do, we get an answer of
18๐ centimetres squared. So great, we can say the area of
the whole sector is 18๐ centimetres squared.

So now, what we can do is actually
move on and find the area of the triangle. Well, in order to find the area of
triangle ๐ด๐๐ต, we actually have a formula that can help us. And that formula is area of
triangle is equal to a half ๐๐ sin ๐ถ. And what this actually tells us is
that the area is gonna be equal to a half multiplied by ๐๐, where ๐๐ are- are
the two sides either side of the angle ๐ถ. So itโs a half ๐๐ multiplied by
sin ๐ถ.

And in our triangle, ๐ and ๐ are
both gonna be 12 centimetres because theyโre the radius and the radius is either
side of 45 degrees, which is ๐ถ our angle in the middle. So therefore, we can now substitute
in the values ๐ equals 12, ๐ equals 12, and ๐ถ equals 45 into our formula. And when we substitute the values
in, we get the area is equal to a half multiplied by 12 multiplied by 12 multiplied
by sin 45. So then, we have this equal to 72
multiplied by root two over two. We get 72 because a half multiplied
by 12 multiplied by 12. So itโs a half multiplied by
144. So a half of 144 is 72. And then, we get root two over two
because this is the exact value of sin 45. And so these are the ones we should
actually remember.

Okay, so weโll calculate this and
when we do, we get 36 root two centimetres squared and thatโs because 72 divided by
two is 36. So we get 36 root two centimetres
squared and thatโs the area of the triangle ๐ด๐๐ต. So now, the next stage is to
actually subtract the area of our triangle from the area of the whole sector to
actually find our area of our shaded section.

So now, if we come back to the
diagram that I drew at the beginning just to help us understand whatโs happening,
weโve got 18๐ because that was the area of the whole sector minus 36 root two
because thatโs the area of the triangle ๐ด๐๐ต which is gonna be equal to 5.6369 et
cetera.

So itโs actually at this point what
we do is we check what the question wants our answer left in. And we can see that it says that it
wants it accurate to three significant figures. So therefore, we can say that the
area of the shaded section is equal to 5.64 centimetres squared to three significant
figures.