Work out the area of the given
shape, giving your answer in terms of 𝜋.
Whilst it may not look like it,
this shape is a sector of a circle. And we know the formula for area of
a sector is a half multiplied by 𝑟 squared multiplied by 𝜃, where 𝑟 is the radius
and 𝜃 is the central angle of the sector in radians. We can use this information to find
the area of our sector, but there are two things we’re going to need to do.
First, we’ll begin by working out
the central angle of our sector. To do this, we use the fact that
angles around a point sum to 360 degrees. We know that this little symbol
means this angle is a right angle; it’s 90 degrees. To find the size of this central
angle then, we’re going to subtract 90 from 360. And that tells us the central angle
270 degrees. However, this formula only works if
the central angle is given in radians. So we’re going to use the fact that
we know that two 𝜋 radians is equal to 360 degrees.
We can find the number of radians
that is equal to one degree by dividing everything through by 360. And when we do, two 𝜋 over 360
simplifies to 𝜋 over 180. So one degree is equal to 𝜋 over
180. We can now find the size of the
central angle of our sector by multiplying the angle in degrees by 𝜋 over 180. That’s 270 multiplied by 𝜋 over
180, which is equal to three 𝜋 over two. Now we probably could have spotted
this. Remember a full turn is equal to
two 𝜋. That means half a turn must be
equal to one 𝜋. We can see that for our central
angle, we could complete half a turn and then a quarter of a turn. That’s 𝜋 plus another half 𝜋,
which we know is the same as three 𝜋 over two.
We now have everything we need to
be able to find the area of our sector. Its radius is five and its angle in
radians is three 𝜋 over two. So the area is a half multiplied by
five squared multiplied by three 𝜋 over two. Five squared is five multiplied by
five, which is 25. And when we’re multiplying
fractions and integers, it’s sensible to give that integer a denominator. And we give it the denominator of
one since 25 is the same as 25 ones. We then multiply the numerators. One multiplied by 25 multiplied by three 𝜋 is 75𝜋. And when we multiply the
denominators, two multiplied by one multiplied by two is four. So the area is 75𝜋 over four or 75
over four 𝜋. There are no units in the question,
so we don’t actually need to give units in the answer. But if we want to, we could say it
is equivalent to 75 over four 𝜋 units squared.