To solve this, we are going to use the formula for the area of the sector of a circle:
![A= \frac{1}{2} r^2 \alpha](https://tex.z-dn.net/?f=A%3D%20%5Cfrac%7B1%7D%7B2%7D%20r%5E2%20%5Calpha%20)
where
![A](https://tex.z-dn.net/?f=A)
is the area of the circular sector.
![r](https://tex.z-dn.net/?f=r)
is the radius of the circle.
![\alpha](https://tex.z-dn.net/?f=%20%5Calpha%20)
is the central angle in radians.
We know form our problem that that the measure of the central angle is 1 radian, so
![\alpha =1](https://tex.z-dn.net/?f=%20%5Calpha%20%3D1)
. We can also infer from the picture that the radius of the circle is 3in, so
![r=3in](https://tex.z-dn.net/?f=r%3D3in)
. Lets replace those values in our formula to find
![A](https://tex.z-dn.net/?f=A)
:
![A= \frac{1}{2} r^2 \alpha](https://tex.z-dn.net/?f=A%3D%20%5Cfrac%7B1%7D%7B2%7D%20r%5E2%20%5Calpha%20)
![A= \frac{1}{2} (3in)^2(1)](https://tex.z-dn.net/?f=A%3D%20%5Cfrac%7B1%7D%7B2%7D%20%283in%29%5E2%281%29)
![A=4.5in^2](https://tex.z-dn.net/?f=A%3D4.5in%5E2)
We can conclude that the area of the circular sector in the picture is
4.5 square inches.
To prove that the arc length is indeed 3 inches, we are going to use the formula:
![A_{L}=r \alpha](https://tex.z-dn.net/?f=A_%7BL%7D%3Dr%20%5Calpha%20)
where
![A_{L}](https://tex.z-dn.net/?f=A_%7BL%7D)
is the arc length.
![r](https://tex.z-dn.net/?f=r)
us the radius of the circle.
![\alpha](https://tex.z-dn.net/?f=%20%5Calpha%20)
is the central angle.
We know from our problem that
![r=3in](https://tex.z-dn.net/?f=r%3D3in)
, and
![\alpha =1](https://tex.z-dn.net/?f=%20%5Calpha%20%3D1)
, so lets replace those values in our formula:
![A_{L}=r \alpha](https://tex.z-dn.net/?f=A_%7BL%7D%3Dr%20%5Calpha%20)
![A_{L}=(3in) \alpha](https://tex.z-dn.net/?f=A_%7BL%7D%3D%283in%29%20%5Calpha%20)
![A_{L}=3in](https://tex.z-dn.net/?f=A_%7BL%7D%3D3in%20)
We can conclude that the length of the arc is indeed
3 inches.