# PSAT Math : How to find the length of a diagonal of a polygon

## Example Questions

### Example Question #105 : Plane Geometry

Regular Octagon has sidelength 1.

Give the length of diagonal .      Explanation:

The trick is to construct segments perpendicular to from and , calling the points of intersection and respectively. Each interior angle of a regular octagon measures ,

and by symmetry, ,

so .

This makes and  triangles.

Since their hypotenuses are sides of the octagon with length 1, then their legs - in particular, and - have length Also, since a rectangle was formed when the perpendiculars were drawn, .

The length of diagonal is .

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

Regular Polygon (a twelve-sided polygon, or dodecagon) has sidelength 1.

Give the length of diagonal to the nearest tenth.      Explanation:

The trick is to construct segments perpendicular to from and , calling the points of intersection and respectively. Each interior angle of a regular dodecagon measures .

Since and are perpendicular to , it can be shown via symmetry that they are also perpendicular to . Therefore, and both measure and and are triangles with long legs and . Since their hypotenuses are sides of the dodecagon and therefore have length 1, .

Also, since Quadrilateral is a rectangle, .

The length of diagonal is . 