What is the length of the side of a square if the apothem is 7

Even though this is a square, it is still technically a polygon. That square has a center.  From that center we can draw 4 radii to each of the pointy parts of the square, creating 4 interior angles. Each one of those interior angles measures 90 degrees, cuz 360 degrees is all the way around the interior angles, ending up where you started off.  So that means that one of those representative triangles has a top vertex of 90. When we drop the apothem from that vertex, which is also the altitude of the triangle, we can cut that representative triangle into 2 right angles.  The altitude cuts not only the angle of 90 degrees in half (45 degrees), it cuts the side it goes through directly in half, too.  That's what we are solving for.  Actually, when we solve that it will only give us half the length of the base of the whole representative trianlge, but we can just multiply whatever length we find by 2 to find the whole base of the trianlge, which of course is the side length we are looking for.  When we pull out one of those right triangles, we have a top angle of 45, another angle of 90, so the third angle has to be a 45 as well (45-45-90) with a height of 7.  We can then use one of our right triangle trig identities to solve for the missing side of the right triangle.  If we use the base angle of 45 and the height of 7 (which is opposite the angle) and we are solving for the base (which is adjacent to the angle) we know that tangent has us covered:



and x=7.  Now remember that that is only half the base of the whole triangle so the base is 7*2 which is 14. 14 is the length of one side of the square.  Well, actually 14 is the length of every side of the square right?

1+1=22×1=22-1=11-1=1duh