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

Example Questions

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

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 #2 : 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.