Question 574813
<pre>
The trapezoid (the US word for the UK word "trapezium") 
is isosceles:

{{{drawing(400,1700/11,-4,40,-13,4,

line(0,0,12,-9),line(12,-9,24,-9), line(24,-9,36,0),line(36,0,0,0),
locate(12,-9,A), locate(24,-9,B), locate(36,2,C),locate(0,2,D),
locate(17,-9,12cm),locate(17,2,36cm),locate(3.4,-5,15cm),locate(29.5,-5,15cm)

 )}}} 

We will draw altitudes AF and BE  (in green) perpendicuilar to DC.  Since
EF = AB = 12cm, and 12cm is {{{1/3}}} of DC which is 36cm, DF amd EC are also
12cm each.

{{{drawing(400,1700/11,-4,40,-13,4,
green(line(12,0,12,-9),line(24,0,24,-9)), locate(12,2,F), locate(24,2,E),
line(0,0,12,-9),line(12,-9,24,-9), line(24,-9,36,0),line(36,0,0,0),
locate(12,-9,A), locate(24,-9,B), locate(36,2,C),locate(0,2,D),
locate(17,-9,12cm),locate(17,2,12cm),locate(3.4,-5,15cm),locate(29.5,-5,15cm),
locate(5,2,12cm),locate(28,2,12cm)
 )}}} 

We can find altitude AF by applying the Pythagorean theorem to right 
triangle ADF:

                    AF² + DF² = AD²
                    AF² + 12² = 15²
                    AF² + 144 = 225
                          AF² = 81
                           AF = 9cm

{{{drawing(400,1700/11,-4,40,-13,4,
green(line(12,0,12,-9),line(24,0,24,-9)), locate(12,2,F), locate(24,2,E),
line(0,0,12,-9),line(12,-9,24,-9), line(24,-9,36,0),line(36,0,0,0),
locate(12,-9,A), locate(24,-9,B), locate(36,2,C),locate(0,2,D),
locate(17,-9,12cm),locate(17,2,12cm),locate(3.4,-5,15cm),locate(29.5,-5,15cm),
locate(5,2,12cm),locate(28,2,12cm),locate(12.3,-3.5,9cm)
 )}}} 


          Area =(Average of parallel sides)·(altitude)

          Area = {{{(AB+DC)/2}}}·AF

          Area = {{{(12cm+36cm)/2}}}·9cm
          
          Area = {{{48cm/2}}}·9cm
            
          Area = 24cm·9cm

          Area = 216cm²

Edwin</pre>