Consider a regular polygon with 10 sides, What is the number of triangles

You can NOT use this formula to find each angle in an irregular polygon. Since each angle of an irregular polygon may be of different size, there is no formula for finding individual angle measures.

Finding SUM and EACH interior angles, with and without formulas:

Find the number of degrees in the sum of the interior angles of a decagon.How many sides does a polygon have if the sum of the interior angles is 1080º?Find the number of degrees in each interior angle of a regular pentagon.Each interior angle of a regular polygon measures 140º. Find the number of sides of the polygon.

Abstract: We consider the number of triangles formed by the intersecting diagonals of a regular polygon. Basic geometry provides a slight overcount, which is corrected by applying a result of Poonen and Rubinstein [1]. The number of triangles is 1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956 for polygons with 3 through 12 sides.

Introduction

If we connect all vertices of a regular N-sided polygon we obtain a figure with

= N (N - 1) / 2 lines. For N=8, the figure is:

Careful counting shows that there are 632 triangles in this eight sided figure.

Derivation

All triangles are formed by the intersection of three diagonals at three different points. There are five arrangements of three diagonals to consider. We classify them based on the number of distinct diagonal endpoints. We will directly count the number of triangles with 3, 4 and 5 endpoints (top three figures). We will count the number of potential triangles with 6 endpoints, then correct for the false triangles. In each of the following five figures, a sample triangle is highlighted.

Three, Four and Five Diagonal Endpoints

 

3 diagonal endpoints. There are 56 such triangles in the figure at left.

The number of triangles formed by diagonals with a total of three endpoints is simply

.

 

4 diagonal endpoints. There are 280 such triangles in the figure at left.

There are

combinations of the four diagonal endpoints. For each set of four endpoints, there are four triangle configurations. Thus there are triangles formed.

 

5 diagonal endpoints There are 280 such triangles in the figure at left.

For each of the N vertices of the polygon, there are four other diagonal endpoints which can be placed on the N-1 remaining locations. Thus there are

triangles formed. This is equal to .

Six diagonal endpoints

The number of potential triangles formed by 6 line segments is

, since there are 6 segment endpoints to be chosen from a pool of N. Often potential triangles are not created by three overlapping line segments because the line segments intersect at a single point. counts both of the following two situations.

 

6 diagonal endpoints, resulting in triangle. There are 16 such triangles in the figure at left.

 

6 diagonal endpoints, false triangle. There are 9 interior intersection points in the figure at left where such false triangles can be formed.

We use a result of [1] to count these false triangles. As in that paper, for a regular N-sided polygon, let am(N) denote the number of interior points other than the center where m diagonals intersect. Surprisingly, only the values m = 2, 3, 4, 5, 6 or 7 may occur. The requisite formulae from [1] are reproduced here:

whereif, 0 otherwise.

If there are K line segments that intersect at one common point, where K>2, there arefalse triangles corresponding to that point. Thus the correction term for false triangles is

where the last term represents the contribution of the center point for even N. The correction is 0 for odd N. The number of triangles formed by line segments with six endpoints on the polygon is then:

Result

The table below summaries the results for. These values were checked through use of a computer program performing an exhaustive search.

NTriangles with 3 diagonal endpointsTriangles with 4 diagonal endpointsTriangles with 5 diagonal endpointsTriangles with 6 diagonal endpointsTotal Number of Triangles31000144400851020503562060300110735140105728785628028016632984504630841302101208401260180240011165132023104624257122201980396079669561328628606435171611297143644004100102856172341545554601501550052593516560728021840774437424176809520309401237653516188161224042840175087340419969155045814027132101745201140193807752038160136200

The sequence formed by the total number of triangles was studied by the late Victor Meally in the 1960's, although it appears he did not find our formula for the N-th term. This is sequence A006600 in the On-Line Encyclopedia of Integer Sequences.

To summarize the final result, the number of triangles generated by intersecting diagonals of an N-regular polygon is:

References

[1] Bjorn Poonen, Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Disc. Math. 11 (1998), 133-156. Note that Theorem 1 has a typographical error: in the second line: 232 should be replaced by 262.

How many triangles are in a polygon with 10 sides?

What is a triangle of 10 sides called?

What is the measure of a regular 10 sided polygon?

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