Polygon Complex

Geometry Level 2

If we want to exactly cover a regular octagon with triangles (of any kind), what is the least number of triangles that we must use?

8 7 5 3 6 4

5 solutions

Worranat Pakornrat
Nov 13, 2015

If we draw lines from one vertex of the octagon to the other vertices, we will form 6 triangles, which is the least number we need.

It is also worthed noticing that we can apply this method to any regular n-gon and the least number of interior triangles will equal to n-2 because starting from a triangle, we have 3 vertices. Adding one more vertex and connecting the dots, we will get a quadrilateral with 2 interior triangles. Adding one more vertex, we will get a pentagon with 3 interior triangles and so on. Therefore, the total interior angles in a n-gon = 180(n-2).

I agree that the answer 6 is correct. But, HOW can we prove that it is the least number possible.

Sachin Sharma - 5 years, 7 months ago

I agree with Pranav: the internal angles in n-gon = 180(n-2), so the least number is then 6 for an octagon.

Worranat Pakornrat - 5 years, 7 months ago

Can you add that explanation into your solution? Thanks!

Calvin Lin Staff - 5 years, 6 months ago

@Calvin Lin – It's done. I hope it's clear to others. ;)

Worranat Pakornrat - 5 years, 6 months ago

If you use the 180*(n-2) then divide by 180 degrees (total in a triangle) you get 6

Calvin Francis - 5 years, 6 months ago

An octagon could be formed with 4 isoseles triangles (with a square hole in the centre)

Bob Dilworth - 5 years, 6 months ago

that's what I came up with, too. Just dive the center square into two identical triangles, and you get the 6 triangle-solution.

Bryan Valverde - 5 years, 6 months ago

Thank you for sharing. Still I prefer to make a whole complete octagon. :)

Worranat Pakornrat - 5 years, 6 months ago

Rather than a hole octagon? :-)

Bob Dilworth - 5 years, 6 months ago

Yeah this isn't worded the best. An octagon is an 8 sided 2 dimensional shape. I'm not sure the interior area needs to be made for it to be considered an octagon. But I agree with the solution in general, but 4 is correct also. Based on the wording of the problem.

Josiah McDonald - 5 years, 6 months ago

That logic is similar to saying that "An annulus is also called a circle, because we can ignore the hole that is contained within". I don't think that makes sense.

Calvin Lin Staff - 5 years, 6 months ago

@Calvin Lin – No, it's like saying you need a circle to make an annulus, which is true.

Brett Walker - 5 years, 6 months ago

@Calvin Lin – Last time I checked an annulus is a REGION made up of two concentric circles. However it's still a circle on the outside of the annulus. Like I said the wording isn't the best.

Josiah McDonald - 5 years, 6 months ago

But in your figure above I can count 8 triangles

Meet Shah - 5 years, 6 months ago

This doesn't prove that n-2 is the least just yet. It is part of the final proof though.

Flewk Isdead - 5 years, 6 months ago

i can make a hollow octagon with 4 triangles connected at the vertices with a square in the middle = bisect the square to get 6 triangles to fill the figure. since the problem doesn't say to cover all inside space - i believe the correct answer is actually 4 to make a regular octogonl

Gary Bowman - 5 years, 6 months ago

Pranav Rao
Nov 15, 2015

I am not sure but we can solve it by noticing that if we want the least number of triangles then the vertices of all the triangles should be vertices of octagon. If this is the case then minimum number of triangles will be the sum of interior angles of octagon divided by sum of interior angle of triangle.

Precisely! That's how to solve it, too.

Worranat Pakornrat - 5 years, 7 months ago

Sadasiva Panicker
Nov 18, 2015

Number of diagonals can be drawn from a vertex of a polygon of n sides = n - 3, Then No. of triangles = n-2. So the least number of triangles need = 8-2 =6

Andy Wright
Nov 24, 2015

There has been a debate as to whether it means an octagonal shape on the outside but with a square hole in the middle (4 triangles), or a completely filled one with no holes (6 triangles). I think the question is clear when it uses the word "cover". That leads you to the 6 triangle solution, and I don't think there is any ambiguity

Suri Sayeed Esha
Nov 22, 2015

No of Triangle= (n-2)=(8-2)=6, as total degree sum=6*180=1080

Polygon Complex

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