NAME DATE BAND

SNUG ANGLESADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

We saw previously that regular polygons can sometimes fit snugly together at a single vertex P, and sometimes not.

Snug Not Snug

1. If you’re only using one type of regular polygon (so, for example, only hexagons), find all regular polygons that can fit snugly at a vertex. Are you certain there are no other ones that would work? How do you know?

2. We also saw that if you allow different types of regular polygons to be used, they can sometimes meet up in a snug way. We showed this one was possible (two

regular 5-gons, and a regular 10-gon).

(a) Algebraically show that two regular 12-gons and a regular 3-gon fit snugly together at a vertex.

(b) Algebraically show that two regular 6-gons and two regular 3-gons fit snugly together at a vertex.

(c) If three regular polygons fit together snugly at a vertex, and one was a 3-gon and one was a 7-gon, what is the third polygon?

(d) Show that if you have three regular polygons, and one of them is a 5-gon and one is a 6-gon, there is no possible third polygon that could fit snugly together with the 5-gon and 6-gon.

3. Use Geogebra to create the three snugly fitting polygons in 2a, 2b, and 2c. (Remember there is a regular polygon tool, which will make your life so much easier!)

4. Below are three polygons that fit snugly at a vertex. Now let’s try to fit a regular polygon into the indicated region/gap. Explain why that is impossible!

5. One interior angle of a regular polygon is between 178o and 179o (it can also equal 178o and 179o). What are all possible number of sides this polygon could have?

6. In problem 2c, we saw a huuuuge-number-of-sides polygon was able to fit snugly with some normal side polygons. Your challenge: find as many combinations of regular polygons to fit snugly together at a vertex. You can use algebra, Geogebra, whatever!