# 3D figure cut by a plane

• Jan 9th 2013, 11:58 PM
3D figure cut by a plane
Question is that there is a cube with base ABCD and top PQRS. A plane cuts this cube and passes through the mid-points of DC,CB and RB. What shape is formed on the Solid's surface. How do I solve such questions? Any good online resource for learning this technique?
• Jan 10th 2013, 12:43 AM
grillage
Re: 3D figure cut by a plane
Have you tried sketching it? Also, the solid has more than one surface so we should be a bit more specific.
• Jan 10th 2013, 01:21 AM
Re: 3D figure cut by a plane
It is a hexagon. How do I prove this mathematically...?????
• Jan 10th 2013, 01:24 AM
Re: 3D figure cut by a plane
• Jan 10th 2013, 03:30 AM
grillage
Re: 3D figure cut by a plane
Not sure what tools you have available to you, but the simplest way is to imagine the side of the cube is 2, then you can take two vectors (1,1,0) and (0,1,-1) and find the angle between them using the dot product. It will come out to 120 degrees and the symmetry should not require you to work out other angles. If you do not have vectors in your locker then you will have to play around with Pythagoras I think.
• Jan 10th 2013, 07:43 AM
Soroban
Re: 3D figure cut by a plane

If the answer is hexagon, there is a typo.
. . Please correct it.

Quote:

There is a cube with base $ABCD$ and top $PQRS.$
A plane cuts this cube and passes through the mid-points of $DC, CB$ and ${\color{blue}RB}.\;\;{\color{red}??}$
What shape is formed on the plane ?

Assuming $P$ is above $A$, $Q$ is above $B$, etc.
. . the diagram looks like this:

Code:

```        P*---------*S         /:        /|         / :      / |       /  :      /  |     Q*---+-----*R  |       |  :    |  |       |  A* - - + - *D       |  /      |  /       | /      | o M       |/        |/     B*----o----*C           N```
$M$ is the midpoint of $DC$; $N$ is the midpoint of $CB.$

But $RB$ is a diagonal of the front face.
The third midpoint is $O$, the center of square $BCRQ$.

The shape is right triangle $MNO.$
• Jan 10th 2013, 07:58 AM
grillage
Re: 3D figure cut by a plane
I also made the assumption initially that P is above A. That would seem logical right? Well looking at the image he attached, it seems that the assumption is wrong. So given the way he has drawn it, we do get a hexagon.
• Jan 10th 2013, 10:15 AM