A diagonal is a line segment joining two non-adjacent vertices of a polygon. A regular hexagon has six sides and six vertices. One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals. Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

where.

A diagonal connects two non-consecutive vertices of a polygon. A hexagon has six sides. There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon. However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals. An octagon has 20 diagonals.

The key is to examinein thie figure below:

Each interior angle of a regular hexagon, including, measures, so it can be easily deduced by way of the Isosceles Triangle Theorem that., so by angle addition,

.

Also, by symmetry,

,

so,

andis atriangle whose long leghas length.

By theTheorem,, which is the hypotenuse of, has lengthtimes that of the long leg, so.

The key is to examinein thie figure below:

Each interior angle of a regular hexagon, including, measures, so it can be easily deduced by way of the Isosceles Triangle Theorem that., so by angle addition,

.

Also, by symmetry,

,

so,

andis atriangle whose hypotenusehas length.

By theTheorem, the long legofhas lengthtimes that of hypotenuse, so.

The key is to examinein thie figure below:

Each interior angle of a regular hexagon, including, measures, so it can be easily deduced by way of the Isosceles Triangle Theorem that. To findwe can subtractfrom. Thus resulting in:

Also, by symmetry,

,

so.

Therefore,is atriangle whose short leghas length.

The hypotenuseof thistriangle measures twice the length of short leg, so.

The key is to examinein thie figure below:

Each interior angle of a regular hexagon, including, measures, so it can be easily deduced by way of the Isosceles Triangle Theorem that. To findwe subtractfrom. Thus resullting in

Also, by symmetry,

,

so,

andis atriangle whose short leghas length.

The long legof thistriangle measurestimes the length of short leg, so.