Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

A cuboid of surface area S is cut by 6 planes as shown. Each plane is parallel to a face, but its distance from the face is random. Now the cuboid is separated in 27 smaller parts.  What in terms of S is the total surface area of all 27 smaller parts?  So first let's just consider one of the planes, right here. This plane ends up cutting out the bottom face of all these top cubes, and the top faces  of all these middle cubes, right here. And in total, those faces contribute to 2 times the area of this top slice, right here.  So if we group them together, this top slice and this plane slice, we can count the top surface area 3 times.  Similarly if we group together the bottom face and this plane, we count the bottom surface area 3 times.  We can do a similar procedure for these groups of planes, as well as these planes. And in the end of the day, we count each face 3 times in this process, so the total surface  area is 3 times S.

Video Summary

The original cuboid is cut into 27 smaller parts by 6 planes, each parallel and randomly distanced from a face. When considering the effect of these cuts, each face of the original cuboid effectively contributes to three separate smaller surfaces due to the intersection of planes. Through this division, each original face ends up being counted three times. Therefore, the total surface area of all the smaller cuboids combined is three times the surface area of the original cuboid. Hence, the total surface area of the 27 parts is 3S.

Keywords

cuboid

planes

surface area

division

geometry