Definition: The diagonal of a polygon is a
line segment
linking two non-adjacent
vertices.

Try this Adjust the number of sides of the polygon below, or drag a vertex
to note the behavior of the diagonals.

A diagonal of a polygon is a line segment joining two vertices. From any given vertex, there is no diagonal to the vertex on either side of it, since that would lay on top of a side. Also, there is obviously no diagonal from a vertex back to itself. This means there are three less diagonals than there are vertices. (diagonals to itself and one either side are not counted).

As described above, the number of diagonals from a single vertex is three less than the the number of vertices or sides, or (n-3).

There are N vertices, which gives us n(n-3) diagonals

But each diagonal has two ends, so this would count each one twice. So as a final step we divide by 2, for the final formula: whereOne of the characteristics of a concave polygon is that some diagonals will lie outside the polygon. In the figure above uncheck the 'regular" checkbox. The drag one of the vertices towards the center of the polygon. You will see white areas appear. The polygon is filled with a yellow color, so what you are seeing is a diagonal that lies outside the concave polygon.

The above formula gives us the number of distinct diagonals - that is, the number of actual line segments. It is easy to miscount the diagonals of a polygon when doing it by eye.

If you glance quickly at the
pentagon on the right, you may be tempted to
say that the number of diagonals is 10. After all, there are two at each vertex, and 5 vertices.
Some people see them making three triangles, for 6 diagonals.
**But there are only 5 diagonals.** Count them carefully.

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

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