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Number of Diagonals in a Polygon - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A diagonal is a segment that connects two non-consecutive vertices in a polygon. The number of diagonals in a polygon that can be drawn from any vertex in a polygon is three less than the number of sides. To find the total number of diagonals in a polygon, multiply the number of diagonals per vertex (n - 3) by the number of vertices, n, and divide by 2 (otherwise each diagonal is counted twice).

How many diagonals are a polygon with 300 sides? Well wouldn't make a whole lot of sense to draw a polygon with 300 sides and draw on all those diagonals. There has to be a shortcut or a formula.
Well, first let's back up. What is a diagonal? A diagonal is any line segment that connects two non-consecutive vertices. So if we look at a triangle. If I look at every single vertex, again the vertex is where two ends meet two sides meet. There's no way for me to draw on a diagonal here because for this vertex both of these sides are consecutive. So there's no way for us to have any diagonals.
If I look at a square however, I can see that there is one non-consecutive vertex if I look at this vertex. I look at another vertex there's only one non-consecutive vertex. So let's see if we can figure out the pattern. To do that, we're going to use this table over here where I have three columns; one for the number of vertices, one for the number of diagonals per vertex and the total number of diagonals that we see in a polygon.
So we've already started with two different polygons. We've talked about a triangle. So, number of vertices in a triangle well, that's just three. The number of diagonals we said was zero because there is no way for us to draw in a diagonal. Which means our total diagonals is still zero. Okay?
Let's go back and look at the square. The square we said, there are 1, 2, 3, 4 vertices. This vertex right here has only one diagonal, this vertex right here has only one diagonal so we're four vertices, each vertex has one diagonal but we only see two of them. So we see that there's going to be some sort of division that's going to have to go on here.
Last, let's look at a pentagon. If I look at this vertex, I can draw in one, two diagonals. And I'm going to see that for every vertex, I'm going to be able to draw in, two different diagonals. So number of vertices here is five, number of diagonals per vertex is two and the total of diagonals here we have a little star so we have five diagonals. So I want to know first for n vertices because I'm going to draw dot dot dot for n vertices, what will be the total number?
Well I see that if I multiply 3 times 0, so we make a dot here. 3 times 0 is 0 so we're okay there. Here we have 4 times 1, but that does not equal 2. So what we're going to have to do is I'm going to have to take 4 times 1 and divide that in half. 5 times 2 divided in half is equal to 5. So I look at the number of vertices we have three so we're going to call that n. Here we have number of diagonals per vertex, here we have 0, 1 and 2 and I see that to get from 3 to 0 I'm going to subtract 3 to get from 5 to 1, I subtract 3 from 5 to 2 I subtract 3. So we have n times the quantity of n minus 3 all divided by 2.
So two key things about this formula right here which tells you the number of diagonals and I'm going to abbreviate DIAG.
So the number of diagonals, there's two key things I want to point out. The first is this n-3. Where is n-3 come from? Well if we have five vertices here. We're not going to count the vertex that's itself because you can't draw a vertex to itself plus there are two more consecutive vertices for total of three vertices in this polygon that we're not really counting.
Second key part here is this divide by 2. Why do we have to divide this by 2? If I go back to this to the square, if I look at this vertex, I've drawn in one diagonal. From this vertice's perspective, I've only drawn one. From this vertex's perspective I've drawn another diagonal. But it's the same diagonal. So every vertex and every diagonal that we draw from a vertex will be counted twice which is why we have to divide our formula by two.

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