The area of a circular sector is
561.3 centimetres squared and the central angle is 27 degrees. Find the radius of the circle
giving the answer to the nearest centimetre.
We’re told the area of the sector
and the measure of the central angle. Let’s recall the formula that
relates these two. The formula for area of a sector
with radius 𝑟 and angle 𝜃 radians is a half 𝑟 squared 𝜃. So before we can use this formula
to form an equation, we’ll need to convert our angle from degrees into radians. And to do this, we recall that two
𝜋 radians is equal to 360 degrees.
We can divide through by 360. And that tells us that one degree
must be equivalent to two 𝜋 over 360 radians, which simplifies to 𝜋 over 180
radians. This means we can change from
degrees into radians by multiplying by 𝜋 over 180. So 27 degrees is equal to 27
multiplied by 𝜋 over 180 radians, which is equal to three 𝜋 over 20 radians.
Now, we can substitute what we know
into the formula for area of a sector. We don’t yet know the radius of the
circle. So the area is a half multiplied by
𝑟 squared multiplied by three 𝜋 over 20. That is of course equal to
561.3. So we formed an equation for
We’re going to solve this equation
by multiplying both sides by 40. Now we could’ve multiplied by two
and then by 20 to get rid of these two denominators. But multiplying by 40 is slightly
quicker. 561.3 multiplied by 40 is
22452. And then, our next step is to
divide both sides by three 𝜋. And that tells us that 𝑟 squared
is equal to 2382.23 and so on.
We won’t round this number just
yet. Instead, we’re going to find the
square root of both sides of our equation. So we’re going to square root this
number in its unrounded form.
Now, when we find the square root
of a number, we should remind ourselves that there are two solutions, a positive and
a negative. In this case though, we can
immediately disregard the negative solution as our radius is a length. And a length cannot be
negative. So 𝑟 is equal to 48.808 and so
on. Correct to the nearest centimetre,
So the radius of our circle is 49