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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
.
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