To solve this, we are going to use the formula for the area of the sector of a circle: [tex]A= \frac{1}{2} r^2 \alpha [/tex]
where
[tex]A[/tex] is the area of the circular sector.
[tex]r[/tex] is the radius of the circle.
[tex] \alpha [/tex] is the central angle in radians.
We know form our problem that that the measure of the central angle is 1 radian, so [tex] \alpha =1[/tex]. We can also infer from the picture that the radius of the circle is 3in, so [tex]r=3in[/tex]. Lets replace those values in our formula to find [tex]A[/tex]:
[tex]A= \frac{1}{2} r^2 \alpha [/tex]
[tex]A= \frac{1}{2} (3in)^2(1)[/tex]
[tex]A=4.5in^2[/tex]
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: [tex]A_{L}=r \alpha [/tex]
where
[tex]A_{L}[/tex] is the arc length.
[tex]r[/tex] us the radius of the circle.
[tex] \alpha [/tex] is the central angle.
We know from our problem that [tex]r=3in[/tex], and [tex] \alpha =1[/tex], so lets replace those values in our formula:
[tex]A_{L}=r \alpha [/tex]
[tex]A_{L}=(3in) \alpha [/tex]
[tex]A_{L}=3in [/tex]
We can conclude that the length of the arc is indeed 3 inches.
