# ACT 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

Polygon  is a regular nine-sided polygon, or nonagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid.

and . being angles of a nine-sided regular polygon, have measure

The other two angles are supplementary to these:

The length of one side of the nonagon is one-ninth of 500, so

The trapezoid formed is below (figure NOT drawn to scale):

Altitudes  and  to the base have been drawn, so

This makes 140 the best choice.

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

Polygon  is a regular seven-sided polygon, or heptagon, with perimeter 500. Which choice comes closest to the length of diagonal ?

Explanation:

Congruent sides , and , and the diagonal  form an isosceles trapezoid.

and . being angles of a seven-sided regular polygon, have measure

The other two angles are supplementary to these:

The length of one side is one-seventh of 500, so

The trapezoid formed is below (figure NOT drawn to scale):

Altitudes  and  to the base have been drawn, so

This makes 160 the best choice.