The perimeter of a circular sector
is 36 centimeters and the central angle is 0.4 radians. Find the area of the sector giving
the answer to the nearest square centimeter.
We recall the formula for area of a
sector with radius 𝑟 and angle 𝜃 radians is a half 𝑟 squared 𝜃. We also know the arc length is
𝑟𝜃. This is useful because we’ve been
told the perimeter of the circular sector. The perimeter is the entire way
around the shape. So if we look at the diagram we’ve
drawn, we can see that’s the arc length plus the two radii.
And in general, we can say that the
perimeter of a circular sector can be found by adding two lots of the radii to its
arc length. It’s two 𝑟 plus 𝑟𝜃. In fact, we know that our circular
sector has a perimeter of 36 centimeters. We’re also told that its central
angle is 0.4 radians. We can use this information to form
an equation in terms of 𝑟. Solving this for 𝑟 will allow us
to find the area of our sector.
Since 𝜃 is equal to 0.4, we can say
that the perimeter of our circular sector is two 𝑟 plus 0.4𝑟. And in fact, this must be equal to
36. Simplifying by collecting like
terms on the right-hand side of our equation, we get 2.4𝑟 is equal to 36. And we solve this for 𝑟 by
dividing both sides by 2.4. 36 divided by 2.4 is 15. So we found that the radius of our
circular sector is 15 centimeters.
All that’s left is to substitute
what we know now about the area of our sector into the formula. It’s a half multiplied by 15
squared multiplied by 0.4, which is equal to 45.
So the area of our sector is 45