Three mutually tangent circles

[Megrs]

Homework Statement

Three mutually tangent circles with the same radius r enclose a shaded area of 24 square units. Determine the value of r to the nearest unit.

Homework Equations

do i use the arc length formula to find the answer?


3. The Attempt at a Solution

A=(central angle*pi*r^2)/360
the central angle is 60 because if u draw a triangle between the three centers of the circles it is equilateral...but i don't know how to get r with only this information

 

[tiny-tim]

Hi Megrs@!

(have a pi: π and a degree: ° and try using the X² tag just above the Reply box )

Three mutually tangent circles with the same radius r enclose a shaded area of 24 square units. Determine the value of r to the nearest unit.
…
the central angle is 60 because if u draw a triangle between the three centers of the circles it is equilateral...but i don't know how to get r with only this information

You're almost there!

Stay with that triangle …

it has the shaded area inside it, so what's left?

 

[Megrs]

Hi Megrs@!

(have a pi: π and a degree: ° and try using the X² tag just above the Reply box )


You're almost there!

Stay with that triangle …

it has the shaded area inside it, so what's left?

what is left are the sectors of the circles and if u make each one=x then 3x+24=A of the triangle. the arc length is r(2*pi*central angle/360) i don't think that gives any useful information though...except maybe help prove the answer once i find it...

 

[tiny-tim]

Hi Megrs!

(what happened to that π i gave you? )

Yes, three sectors …

now, you know the radius and the angle of each sector, so the area is … ?

 

[Megrs]

Hi Megrs!

(what happened to that π i gave you? )

Yes, three sectors …

now, you know the radius and the angle of each sector, so the area is … ?

what n?? i don't get it...
no, i know the angle of the sectors...i do NOT know the radius that is what i have to find...so i don't know the area of the sectors either

 

[tiny-tim]

(it's not an "n", its a "π" !)

Yes, but you know that r is half the side of your equilateral triangle! …

so the area of the shaded part (as a multiple of r) is … ? ​

 

[Megrs]

i know r is half the side i don't get how to relate this all together because the shaded part is not part of the circles...the area of the shaded part is 24. the area of a sector is (24-area of the triangle)/3...

 

[tiny-tim]

so as a factor of r, the area of a sector is … ?

 

[Megrs]

so as a factor of r, the area of a sector is … ?

area of a sector is 60*r^2*pi/360

 

[tiny-tim]

area of a sector is 60*r^2*pi/360

so as a multiple of r, the area of the shaded part is … ?

 

[Megrs]

i don't know because the shaded part is not part of the sectors...
r=sqrt(360*area of sector/60pi)
?