Geometry Worksheet 150 - Platonic Views

We had a little trouble answering questions #1 & #2. Hope you can help.

Question 1 asks how many of the 11 polygons are regular. The answer says seven.

First my son came up with six: #4 hexagon, #5 quadrilateral, #7 triangle, #9 decagon, #10 decagon, #11 hexagon

Upon further reflection he realized that for there to be 11 polygons, he had to include the pentagon in #9 and the triangle in #11. Since these are regular, he then came up with the answer eight.

Question #2 asks you to name the congruent regular polygons. The answer is decagon.

My son's first answer was: hexagon, quadrilateral, triangle, pentagon and decagon (ignoring duplicate shapes). In reviewing this question with him, he realized he missed the word 'congruent' in the question. So he revised his answer to decagon and hexagon (since there was only one pentagon and one quadrilateral, and the triangles were different sizes). I told him that decagon was the only answer. So he measured the hexagons and discovered #11 was slightly smaller (but in an irregular way - the sides are not all the same length). So he changed his answer to decagon only.

He also went back to question #1 and discarded the hexagon in #11 because it is not regular, so then his answer to question #1 was seven (same as the solution page).

This all makes sense to me, except for the irregular hexagon in #11. How can he demonstrate the view of a 20-sided platonic solid (icosahedron) with an irregular polygon outline? My son says the outline of the view doesn't need to be equilateral, because it's just an outline. In other words, if he were demonstrating a face of the icosahedron, then it would need to be equilateral. I say if you are demonstrating a view of a Platonic solid which is made up of regular polygons with identical vertices, then anys views of a Platonic solid would have to be drawn with regular polygons.

Is his rationale correct and am I off track? Was the hexagon in #11 deliberately drawn with unequal sides for the purpose of question #2? Are my son's choices for questions #1 and #2 above correct?

Thanks for your help,
Julia & Liam

Dear Julia and Liam,


Wow, your questions took some thinking (and revising). The answer to #1 should be 8; #11 was supposed to be a regular hexagon. It is now. But now #2 isn't right. There are two answers: the hexagon (size of #4 is now changed to match #11) and the decagon.


As far as your question about the outlines of Platonic solids always giving a polygon. It seems to make sense that they should. But look at the outlines for the tetrahedron and cube in #3, #6, and #8. They aren't regular.


The changes will be on the Correction page soon. Thanks.


Joan A. Cotter

Dear Joan,

Thank you for clearing up our questions. It all makes sense now. I even understand the part about the outlines now.

Thanks,
Julia & Liam