SOLUTION: The figure shows four circular pipes, all with 12-inch diameters, that are strapped together with a snugly-fitting band. How long is the band?

Make a sketch.


From the sketch, find 4 linear elements of the band, for which the length of each element is equal to 12 inches.


Find also four arcs of 90 degrees each with the radius of 12/2 = 6 inches.


The sum of the lengths of these elements is   =  = 48 + 2*3.14*6 = 85.68 inches, approximately.


ANSWER.  The band length is   = 48 + 2*3.14*6 = 85.68 inches, approximately.

    The first step of the solution is to make a sketch.

    After that, the rest of the solution is OBVIOUS.

    The new band total length will be the same.

    Edwin, feel free to develop your solution (!)

    Have a nice day and be safe (!)

You did not draw the figure.  Ikleyn assumed the pipes were tied like this:



Most people have had the experience of putting a rubber band tight around 4
identical pens, and have seen that they kind of shift on their own like the
figure below.  So I believe the figure below was what you were given, right?  



I'll draw it bigger and put in some lines in red.



The four quadrilaterals which appear
to be rectangles indeed ARE rectangles because radii drawn
to the points of tangency are perpendicular to the tangent.
The interior angles of a rectangle are 90° each.

Therefore the 4 straight parts of the band are 2 radii each or
1 diameter or 12 inches,  So the straight parts of the band 
are 4∙12  = 48 inches. 

The triangle connecting the centers of any three mutually 
tangent circles is an equilateral triangle because the sides
are two radii or 1 diameter or 12 in. each.  The interior angles
of an equilateral triangle are 60° each.

So the 60° and 90° angles are correct as indicated in the 
figure.   

The circumference of any one of the 4 circles is π∙d or 
12π inches.

Adding the four marked angles in the upper left circle and 
subtracting from 360° gives 60° for each of the 2 shorter 
curved parts of the band.  Since 60° is 1/6 of 360°, each 
shorter curved part of the band is 1/6th of 12π or 2π 
inches  Therefore the 2 shorter curved parts of the band are 
4π inches.

Adding the three marked angles in the upper right circle and 
subtracting from 360° gives 120° for each of the 2 longer 
curved parts of the band.  Since 120° is 1/3 of 360°, each 
longer curved part of the band is 1/3th of 12π or 4π  
Therefore the 2 longer curved parts of the band are 8π
inches.

So the sum of all the curved parts is 4π+8π = 12π inches.
Notice that they add up to one of the circle's circumference.

Adding the straight parts and the curved parts, we get

length of band = 48 + 12π inches, about 85.68 inches.

So Ikleyn was right that the band has the same length whether
the pipes are arranged as Ikleyn assumed them to be or as I 
assumed them to be.    

Edwin