The area of a circular sector is
1888 square centimetres and its central angle is 1.7 radians. Find the arc length of the sector
giving the answer to the nearest centimeter.
In this problem then, we have a
circular sector. We know its area is 1888 square
centimeters and we know its central angle, which we can call 𝜃, is 1.7 radians. We want to calculate the arc length
of the sector, which we often denote using the letter 𝑠. The key thing to note is that the
central angle of the sector is measured in radians. So the formulae that we’re going to
use for both the area of the sector and the arc length must be the formulae in
radians. Those are a half 𝑟 squared 𝜃 for
the area of the sector and 𝑟𝜃 for the arc length.
In order to calculate the arc
length then, we need to know both the radius and the central angle. Let’s use the information we were
given about the area to form an equation. The area is 1888 square
centimeters. We don’t know the radius, so we’ll
keep the letter 𝑟. And central angle 𝜃 in radians is
1.7. We can now solve this equation to
work out the radius of the sector, which we’ll then be able to substitute into our
formula for the arc length. We can multiply both sides of the
equation by two and then divide by 1.7 giving 𝑟 squared is equal to 1888 multiplied
by two over 1.7, which as a decimal is 2221.176.
To solve for 𝑟, we take the square
root of each side of this equation, taking only the positive value as 𝑟 represents
a length. That gives 47.129, and we’ll keep
this exact value on our calculator display for now. So we’ve now calculated the radius
of the sector, and all that remains is to calculate the arc length. Using the formula arc length equals
𝑟𝜃, we multiply the value we’ve just calculated by the central angle of 1.7, which
gives 80.119. We were asked to give an answer to
the nearest centimeter, so we round down. Using the known area and the known
central angle of this circular sector then, we’ve calculated that the arc length to
the nearest centimeter is 80 centimeters.