Grades 7-8 Video Solutions 2025-2025_7-8

Video Transcription

Problem number four. The regular hexagon shown is divided into many triangles of equal area. What fraction of the hexagon is shaded?  To solve this problem, let's draw the hexagon bigger and notice that in this red triangle, we can count that there are two shaded triangles out of six total.  In particular, you can imagine this as a sort of slice of the hexagon. And if you count all the six slices of the hexagon, you will notice that each of them has the same pattern,  where they have two shaded slices and four unshaded slices. So in total, each of these six sides of the hexagon contributes two shaded triangles and six total triangles.  That is, there are two times six or 12 yellow shaded triangles and six times six or 36 total triangles.  So this means that the fraction that we are looking for is 12 divided by 36 or 1 over 3. So the answer is B, 1 third.

Video Summary

The problem involves determining the fraction of a regular hexagon that is shaded, given it's divided into triangles of equal area. Observing a single slice of the hexagon shows two shaded triangles out of six. Each of the six slices maintains this pattern. Therefore, the entire hexagon consists of 12 shaded triangles out of 36 total triangles. Consequently, the fraction of the hexagon that is shaded is \( \frac{12}{36} \), simplifying to \( \frac{1}{3} \). So, the shaded fraction of the hexagon is 1/3.

Keywords

hexagon

shaded fraction

triangles

equal area

geometry