arc lengthareacirclestrigonometry
The question doesn't state whether its subtended at the center or circumference, but I not sure if it matters
The sector a circle subtended by an angle of $22.5$ degrees has an area of $\frac{9\pi}{4}$ squared meters.
I have never done one of these problems before but after searching the internet I found the formula
$$s=r\theta$$
though this is the formula for the arc length and I'm not sure if you can find the diameter from this formula.
Here are some equal angle inscribed angles (they all subtend half the angle of the red arc). It is clear that the area of the last sector is properly contained in the areas of the others.
$\hspace{2.5cm}$
The area of the sector is the sum of the area between the arc and its chord (a constant area) and the area of the triangle, which is $\frac12$ the product of the length of the chord (a constant length) and the altitude of the triangle, which varies as the vertex moves around the circle.
The perimeter of the sector includes the length of the radius $\times 2$, as well as the arc length. So the perimeter is the length "around" the entire sector, the length "around" a slice of pizza, which includes it's edges and its curved arc.
The arc length is just the curved portion of the circumference, the sector permimeter is the length of line $\overline{AC} = r$ plus the length of line $\overline{BC} = r$, plus the length of the arc ${AOC}$.
The circumference of the circle is the total arc length of the circle.
Length is one-dimensional, the length of a line wrapped around the circle. Area is two dimensional; All of what's inside the circle.
Best Answer
The area of a circular sector is the circle's area times the ratio of the angle and $360^o$.
In this case, that ratio is $22.5^o/360^o=1/16$.
Therefore the area of the circle is $16\times\dfrac{9\pi}4 $m$^2 = 36\pi$ m$^2.$
The area of a circle is $\pi r^2$, so in this case $r^2=36$ m$^2,$ so $r=6$ m.
The diameter is twice the radius: $d=2r=2\times6 $ m $ =12$ m.