𝐴𝐵𝑂 is a sector of a circle with
center 𝑂 and radius 14 centimeters. Calculate the shaded area. Give your answer to one decimal
Now, in order to actually work out
the area of the sector, so our shaded area, what we need to do first is actually
find out what the angle, I put here as 𝑥, is because we’ll need that when we’re
trying to calculate the area using the formula. We’ve actually got this relation
that we know, which is that the angles around a point equals 360.
So we can use this to actually
calculate what 𝑥 is gonna be because this tells us that if we have 𝑥 and we add it
to 126, the answer is gonna be equal to 360 degrees. And therefore, if we actually
subtract 126 from each side, what we’re gonna get is that 𝑥 is equal to 234
degrees. And now, I’ve added this to the
diagram. So this is actually the angle of
Okay, so now, how we’re actually
gonna work out what the area of the sector is going to be. Well, to work out the area of the
sector, what we have is a formula to help us. And that formula is that the area
of the sector is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared. So we actually get that formula
because 𝜃 over 360 tells us the fraction of the circle. So it actually tells us what our
sector is as a fraction of the whole entire circle. And 𝜋𝑟 squared is actually the
area of the circle.
So what we’re saying is that the
fraction of the circle multiplied by the area of the whole circle. And that will give us the area of
this sector that we’re interested in. So therefore, what we’re gonna get
is that the area of our sector is gonna be equal to 234 over 360. And that’s 234 because that’s the
angle of our sector. So it’s a major sector in this case
multiplied by 𝜋 multiplied by 14 squared.
And then, in next stage, if we’re
gonna simplify, if we actually divide 234 and 360 by 18, we get 13 over 20. And then, this is multiplied by 14
squared which is 196. Then, multiply it by 𝜋. So we get 13 multiplied by 196 over
20𝜋, which when calculated will give us 400.2389 et cetera.
So okay, we’ve actually worked out
the area of our shaded sector, but have we finished there? Well, if we check back to the
question, what we want is actually our answer left to one decimal place. So therefore, we can say that the
area of the shaded sector is gonna be equal to 400.2 centimeters squared to one
And we get 400.2 because if we look
back, if we want to one decimal place, I’ve put a line to the right of the one
decimal place because that’s the two. And then, if we look at the number
after it, that’s a three. And because that’s less than five,
we’ll actually keep the two as it is without rounding up. So we’re left with 400.2
centimeters squared to one decimal place.