Video Transcript
An arc has a measure of 63 degrees
and a radius of four. Work out the length of the arc. Give your answer in terms of 𝜋 and
in its simplest form.
For an arc with a radius 𝑟 and the
angle 𝜃 radians, the arc length is 𝑟 multiplied by 𝜃. We have indeed been given the angle
and the radius for our sector, but the angles in degrees. So let’s recall how to change from
degrees into radians.
Two 𝜋 radians is equivalent to 360
degrees. We can scale down by dividing both
sides by 360. And we can see that one degree is
equal to two 𝜋 over 360 radians. That’s equivalent to 𝜋 over
180. This means we can convert 63
degrees into radians by multiplying it by 𝜋 over 180.
Before we do the multiplication,
though, notice that both 63 and 180 have a common factor of nine. 63 divided by nine is seven. And 180 divided by nine is 20. That means that 63 degrees is
equivalent to seven 𝜋 over 20 radians.
We now know the radius of our
sector and the angle 𝜃. So we can substitute this into the
formula for arc length. 𝑟 multiplied by 𝜃 is four
multiplied by seven 𝜋 over 20. Once again, before we actually
perform the calculation, spot that both four and 20 have a common factor of
four. One multiplied by seven 𝜋 over
five is seven 𝜋 over five. And the length of the arc given is
seven 𝜋 over five.
Work out the area of the
sector. Give your answer in terms of 𝜋 and
in its simplest form.
Once again, when we have a sector
with a radius 𝑟 and an angle 𝜃 radians, the sector area is a half multiplied by 𝑟
squared multiplied by 𝜃. We already know the radius of this
sector and the angle measured in 𝜃. So we can substitute these into the
formula. And it gives us one-half multiplied
by four squared multiplied by seven 𝜋 over 20.
We can cross cancel by dividing
four squared and 20 by four. We can then also cross cancel by
dividing four and two by two. One multiplied by two multiplied by
seven 𝜋 is 14𝜋. And then the denominator of this
fraction is five.
The area of the sector is 14𝜋 over
five.
