# Find the object from its shadows

A 3D object is suspended in the air and it casts shadows on each orthogonal plane. The shadows are in the form of a circle, triangle and square, as shown in the diagram below. What is the shape of the mystery object?

Assuming the object we seek is solid all the way through (in the same way as a cube, tetrahedron or sphere would be), this mystery object can be created by...

...intersecting a cylinder with a triangular prism.

The resulting object looks like this:

(Images taken from missouristate.edu).

The circular shadow is formed by shining a light directly along the cylindrical component, the triangular shadow is formed by light hitting the end of the triangular prism head-on, and the square shadow is formed by light hitting the triangular prism from the side.

A nice image showing the shadow effect in action can be seen here:

Jaap Scherphuis makes some good additional points in comments below this post...

Namely that this is not the only object that satisfies this property, and others with smaller volumes can be constructed - for instance, by laying a horizontal circular base board and positioning two intersecting vertical boards - one triangular and one square - on top.

• That's correct and incredibly fast! Jul 3, 2023 at 12:29
• @DmitryKamenetsky it is a bit known 3d graph :) that could be the reason. but still fast solution!
– Oray
Jul 3, 2023 at 12:33
• @DmitryKamenetsky I think I came across this in a book once before! Something clicked immediately when I saw your post...
– Stiv
Jul 3, 2023 at 12:34
• Strictly speaking this answer gives only the largest object with those three shadows. You could cut away quite a lot of the volume without affecting those shadows, until you are left with three thin flat boards - a horizontal circular base board and a two vertical intersecting triangle and square boards on top. There are also other ways to cut a way that volume. Jul 3, 2023 at 13:04
• @DmitryKamenetsky I've worked out where I know this from - it's in another form: an object fitting into three differently shaped holes rather than casting 3 different shadows, but it's the same solution ultimately, just a different scenario. It's Puzzle 70 'The Peculiar Peg' in Alex Bellos's book, So You Think You've Got Problems? (2019)
– Stiv
Jul 3, 2023 at 19:29