Question

A sector of a circle has an arc length of π cm and a central angle of π/6 radians. What is the area of the sector?A. 3π cm²B. 2π cm²C. π cm²D. 5π cm²

Answer 1

Our problem involves the properties and formulas involving a sector.

A sector is a part of a circle it is like a slice of a pie or a cake.

To find the area of a sector we must have its central angle and radius. But in our problem, the central angle is given but there is no sign of radius being given. So first we have to find the radius of our circle.

To find the radius of our circle we can use the given Arc length of the circle. An Arc Length of a Circle (S) is given by the formula:

$S=\theta r$

Where S is the arc length, θ is the Central angle in radians, and r is the raduis. Since we already have a value for S and θ. We can now find the value of r, or the radius.

$\begin{gathered} S=\theta r \\ \pi=\frac{\pi}{6}(r) \\ 6\pi=\pi r \\ 6=r \end{gathered}$

Therefore we now know that the radius of the circle is 6 cm.

Now that we know the radius of the circle we can now find its area using the formula:

$A=\frac{\theta r^2}{2}$

Where A is the area, θ is the central Angle, and r is the radius.

$\begin{gathered} A=^{}\frac{\theta r^2}{2} \\ A=\frac{(\frac{\pi}{6})(6)^2}{2} \\ A=\frac{\pi6}{2} \\ A=3\pi \end{gathered}$

Therefore the answer is 3π cm². Which is OPTION A.