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# For a square there are four, for a regular pentagon there are 11, for a regular hexagon there are 24 and for a regular heptagon there are 50. What are they and how many are there for a regular octagon?

Question #121967. Asked by gmackematix. (Jun 16 11 7:56 PM)

queproblema

Interior triangles when all possible diagonals are drawn.

A regular octagon has 80.

http://en.wikipedia.org/wiki/7-simplex
http://www.jimloy.com/geometry/pentagon.htm
Etc.

 Jun 16 11, 9:15 PM
Watchkeeper

The answer is indeed 80, but the internal shapes are not necessarily triangles. The pentagon, for example, has another pentagon created inside it by the diagonals and an octagon has a hexagon at the centre.

The equation for calculating the number of internal shapes is horrendous:

http://mathworld.wolfram.com/RegularPolygonDivisionbyDiagonals.html

Fortunately, the author of the Webpage lists the numbers as being 1 (triangle), 4 (square), 11 (pentagon), 24 (hexagon), 50 (heptagon), 80 (octagon), 154 (nonagon), 220 (decagon) and 375 (hendecagon).

 Jun 17 11, 5:20 AM
gmackematix

Yay, you have between you, QP and Watchkeeper, correctly joined all the dots. They are the number of separate areas a polygon is divided into by its diagonals, which for a regular octagon is 80.

I think the formula is correct and shows that, just because a problem is easy to state doesn't mean the maths will be simple (see the circumference of an ellipse).

 Jun 17 11, 9:40 PM

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