Question

The area of a 60-degree sector of a circle is 36π cm^2. What is the diameter of the circle? Explain too, please.

Answer 1

Answer:

Unsimplified:  $2\sqrt{216}$ cm

Simplified: $12\sqrt{6}$ cm

Rounded to nearest tenths:  29.4 cm

Rounded to nearest hundredths: 29.39 cm

Rounded to nearest thousandths:  29.394 cm

Rounded to nearest ten-thousandths: 29.3939 cm

Step-by-step explanation:

The area of a circle with radius,$r$, is $A=\pi r^2$.

Since we have a 60 degree sector with radius $r$ then we have  $\frac{60}{360}$ of the area of a circle with radius $r$.

That is we have the following for the area of a 60 degree sector:

$\frac{60}{360} \cdot \pi r^2$

Reduce the fraction:

$\frac{1}{6} \cdot \pi r^2$

We have that this equals $36\pi=\frac{1}{6}\pi r^2$.

This implies $36=\frac{1}{6}r^2$.

Multiply both sides by 6:

$216=r^2$

If you take the square root of both sides, you get:

$\pm \sqrt{216}=r$

The radius only makes sense to be positive so we can throw the negative out.

$r=\sqrt{216}$.

So the radius is $\sqrt{216}$.

The diameter is twice the radius.  Our diameter is therefore $2(\sqrt{216})=2\sqrt{216}$.

We should really go ahead and attach the units.

The answer is $2\sqrt{216}$ cm.

We can simplify this.

$2\sqrt{36}\sqrt{6}$

$2(6)\sqrt{6}$

$12\sqrt{6}$