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Thus for each of the 8 vertices you can draw 5 diagonals and hence you have constructed 5 8 = 40 diagonals. But you have constructed each diagonal twice, once from each of its ends. Thus there are 20 diagonals in a regular octagon.

mathmate

For each vertex of an octagon, we can draw a line to each of the 7 other vertices. Out of the 7, 2 are to adjacent vertices, so they are not diagonals.

That leaves us with 5 diagonal for each of the vertices, giving a total of 5*8=40 diagonals.

However, diagonals are not considered directional, i.e. drawing from each end is the same diagonal, so we need to reduce the number of diagonal by half. The total number of diagonals is therefore 40/2=20.

In the attached diagram, we see the five diagonals drawn from vertex a.

View image mathmate

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