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Example Questions
Example Question #421 : Geometry
Regular Octagon has sidelength 1.
Give the length of diagonal .
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.
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.