Question number 28. The diagram shows a polygon whose vertices are the midpoints of the edges of the cube. An interior angle here of this polygon is defined in the normal way, the angle between the two edges meeting at a vertex. What is the sum of the measures of all the interior angles of the polygon? So what we can imagine here is happening, I will highlight this in the smaller diagram. In blue we have these triangles here, let's say this one and this one, that are straddling a corner of the cube. And the opposite triangle here of that corner would be this one over here, like that. And so we have pairs of these and we can count the sum of the angles here inside of a triangle like that, 180 degrees, for each triangle like this that we find, that would be by symmetry the sum of the interior angles of this polygon. So how many triangles do we have like that? Let's connect all the vertices here that are missing, so I would need to fill in this edge over here, then this edge over here, and opposite of that is this triangle, opposite of the triangle here in the lower left-hand corner would be this triangle here, and then I connect the vertices here for the remaining two and what I get is this hexagon. So here we have a hexagon and for each vertex we count one complete triangle. So we have a total of six, that's the number of vertices, times 180 degrees per a complete triangle or that equals to a thousand and eighty degrees as our total sum. So that's a quick way to do this problem, an equivalent way with more details is available in the suggested solutions, so you can review that also, but the answer remains 180 degrees and that is choice B.

interior angles

polygon

cube

hexagon

geometry