# SAT Math : How to find the length of the diagonal of a hexagon

## Example Questions

### Example Question #1 : How To Find The Length Of The Diagonal Of A Hexagon

How many diagonals are there in a regular hexagon?

Explanation:

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 .

### Example Question #3 : How To Find The Length Of The Diagonal Of A Hexagon

How many diagonals are there in a regular hexagon?

10

3

9

6

18

9

Explanation:

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.

### Example Question #2 : How To Find The Length Of The Diagonal Of A Hexagon

Hexagon  is a regular hexagon with sides of length 10.  is the midpoint of . To the nearest tenth, give the length of the segment .

Explanation:

Below is the referenced hexagon, with some additional segments constructed.

Note that the segments  and  have been constructed. Along with , they form right triangle  with hypotenuse .

is the midpoint of , so

.

has been divided by drawing the perpendicular from  to the segment and calling the point of intersection .  is a 30-60-90 triangle with hypotenuse , short leg , and long leg , so by the 30-60-90 Triangle Theorem,

and

For the same reason, , so

By the Pythagorean Theorem,

when rounded to the nearest tenth.