Common if shapes are detached!

Geometry Level 3

I have a regular pentagon and a regular octagon, with the same side length.

I create a new figure by gluing along a common edge. How many edges does the new figure have?

The answer is 11.

1 solution

A Former Brilliant Member
Sep 26, 2016

Pentagon has 5 sides and octagone has 8 sides If i placed two shapes on the plane and joint its side each other so the one side is merged in other side of shape, it will be common side for those two shapes... Sides = $(5+8)-2=11$ sides.

I am considering only regular polygons (pentagon and octagon) as I cannot say my answer will be justified for irregular ones.

Suppose one of the polygons has a bigger side length than the other. Then the number of sides formed by joining these two polygons side-wise will be $12$ or $13$ . Two diagrams for help are given below:

12 sides:

13 sides:

Now, suppose both have equal side lengths. Then the number of sides in this case is $11$ and not $12$ as the common side that has formed will be merged in between as the common face of the new shape. This is illustrated in the diagram given below:

Thus, based on this, there are 3 possible answers to this problem (and probably more if regular polygons are not considered).

Tapas Mazumdar - 4 years, 8 months ago

Thanks for specific explanation...!

A Former Brilliant Member - 4 years, 8 months ago

Thanks for putting in the effort to write up the solution. Unfortunately, it is incorrect because:

You actually lose both sides / edges, so the answer is 5+8-2 = 11.

Calvin Lin Staff - 4 years, 8 months ago

Common if shapes are detached!

The answer is 11.

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