# Find the diameter of a circle subtended by an angle

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.

• Note $22.5=360/16$ – J. W. Tanner May 14 at 22:16
• I have never encountered a problem like this, could I get one more hint? – Eric Brown May 14 at 22:24
• The area of a circular sector is the circle's area times the ratio of the angle and $360^o$ – J. W. Tanner May 14 at 22:34
• So the equation should be $\frac{9\pi}{4}=Area * \frac{360}{16}$? – Eric Brown May 14 at 22:49
• $\frac{9\pi}4=Area\; of circle \times \frac {22.5}{360}$ – J. W. Tanner May 14 at 22:56

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.