SOLUTION: A metal band is wrapped tightly around pipes of radius 3cm and 9cm. What is the length of the band? Express your answer in simplest radical form.

Extend the two tangent lines until they meet left of the smaller circle.
Draw a line from that point to the center of the larger circle.
Draw in a radius in each circle to the points of tangency:



We have two similar right triangles.
We let h be the length of the horizontal part that extends left of the
small circle.

By ratios of corresponding sides of similar right triangles,


 
Cross-multiply

Divide both sides by 3





Now we know the lengths of the two hypotenuses:

The hypotenuse of the small right triangle = h+3 or 3+3=6.
The hypotenuse of the large right triangle = h+3+3+9 or h+15 or 3+15 = 18.

The upper straight part of the band equals:
the longer leg of the large right triangle MINUS
the longer leg of the small right triangle.

Since the short leg in each triangle is half of the hypotenuse, we
know that both right triangles are 30-60-90 right triangles and that
means the longer leg is the short leg multiplied by .

The long leg of the large right triangle is .
The long leg of the small right triangle is .
So the upper straight part of the band is  -  = .
The lower straight part of the band is also .
So both straight parts of the band are . 

Now we only need to find the curved parts of the band.
We find the curved part around the small circle.
Since the triangle is a 30-60-90 right triangles, the
upper curved part is subtended by a 60-degree angle.
So it is 60/360 or 1/6 of the circumference of the
entire small circle.  Using C=2πr = 2π(3) = 6π.
And 1/6 of that is π.
The lower curved part of the band is also π,
So the curved part of the band around the small
circle is 2π

We find the curved part around the large circle.
Since the triangle is 30-60-90 right triangles, the
upper curved part is subtended by a (180-60) degree angle,
or a 120-degree angle.
So it is 120/360 or 1/3 of the circumference of the
entire large circle.  Using C=2πr = 2π(9) = 18π.
And 1/3 of that is 6π.
The lower curved part of the band is also 6π,
So the curved part of the band around the large
circle is 12π

Adding the curved parts of the band, we get 2π+12π=14π

Adding the curved and straight parts of the band, we get

  <-- answer

Edwin