Hello there. To solve this question, we'll have to remember some properties about circular sectors.
Given a circular sector of a circle with radius R:
Suppose the length of the arc AB is alpha (in radians).
From a well-known theorem about angles in a circle, we know that the angle generating this arc from the center has the same measure, that is:
So we want to determine the area of the sector knowing the radius and the length of the arc.
First, we know that the area of the full circle is given by:
[tex]A=\pi\cdot R^2[/tex]
The sector is a fraction of this circle, that means that:
[tex]A=kA_{sector}[/tex]
A is a multiple of Asector.
In fact, this proportionality constant is the ratio between the central angle and the angle alpha forming the sector, that is
[tex]k=\dfrac{2\pi}{\alpha}[/tex]
It is also possible to have alpha in degrees, but we have to convert the center angle to degrees as well, so we get
[tex]k=\dfrac{2\pi\cdot\dfrac{180^{\circ}}{\pi}}{\alpha\cdot\dfrac{180^{\circ}}{\pi}}=\dfrac{360^{\circ}}{\alpha^{\circ}}[/tex]
As we want to solve for the area of the sector, we have that:
[tex]A_{sector}=A\cdot\dfrac{\alpha^{\circ}}{{360^{\circ}}}=\dfrac{\alpha^{\circ}}{360^{\circ}}\cdot\pi R^2[/tex]
Okay. With this, we can solve the question.
We have the following circle:
In this case, notice R = 18 yd and the length of the arc is 80º. This gives us the angle alpha:
Now, we take the ratio between the angle and the total angle applying the formula:
[tex]A_{sector}=\dfrac{80^{\circ}}{360^{\circ}}\cdot\pi\cdot18^2[/tex]
Square the number
[tex]A_{sector}=\dfrac{80^{\circ}}{360^{\circ}}\cdot\pi\cdot324[/tex]
Simplify the fraction by a factor of 40º
[tex]A_{sector}=\dfrac{2}{9}\cdot\pi\cdot324[/tex]
Multiply the numbers and simplify the fraction
[tex]A_{sector}=\dfrac{648\pi}{9}=72\pi[/tex]
This is the area of this sector.
