Please help me solve this question about circles (sectors).

[puggyycorn]

The question is: The arc length of a sector of a circle is 2π cm and the area of the sector is 6π cm2 . Find the radius of the circle.

Multiple choice: A) 4.5 cm B) 7.5 cm C) 3 cm D) 6cm

I know I can solve this by substituting the answers but I just want to know if there is a way to figure out the answer without substituting the choices. Thanks!

 

[d.mehdoi]

First of all I am a beginner in Maths so my answer might be wrong.

I found a website and it shows us how we can find this information.

Area of a sector of a circle - Math Open Reference

www.mathopenref.com

(I suggest you to see "if we know the arc length" there)

According to this website, when we know the arc length, we can find the area of a sector by (Radius times arc length)/2

As we know the area of the sector here, after I type 3.14 for Pi it is:
18.84=(Radius times 6.28)/2
37.68=Radius times 6.28
Radius=6

But as I said, I am a beginner and this answer might be wrong.

 

[MarkFL]

The question is: The arc length of a sector of a circle is 2π cm and the area of the sector is 6π cm2 . Find the radius of the circle.

Multiple choice: A) 4.5 cm B) 7.5 cm C) 3 cm D) 6cm

I know I can solve this by substituting the answers but I just want to know if there is a way to figure out the answer without substituting the choices. Thanks!

I would begin with the formula for the area of a circular sector:

[MATH]A=\frac{1}{2}r^2\theta[/MATH]
Now, let's arrange the formula for the arc-length as follows:

[MATH]\theta=\frac{s}{r}[/MATH]
Hence:

[MATH]A=\frac{1}{2}r^2\cdot\frac{s}{r}=\frac{1}{2}rs\implies r=\frac{2A}{s}[/MATH]
And then, plug in the given data:

[MATH]r=\frac{2\left(6\pi\text{ cm}^2\right)}{2\pi\text{ cm}}=6\text{ cm}[/MATH]
This is essentially the same thing d.mehdoi did, but without any need to approximate \(\pi\).