Angle trisection is the division of an arbitraryangle into three equal angles. It was one of the three geometric
problems of antiquity for which solutions using only compass
and straightedge were sought (Heath 1921, pp. 235-244).
The compass opening may change between construction steps.
The problem was algebraically proved impossible by Wantzel (1837).
Although trisection is not possible for a general angle using a Greek construction, there are some specific angles,
such as
and radians ( and , respectively), which can be trisected. Furthermore,
some angles are geometrically trisectable, but cannot be
constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary
anglecan be accomplished using a markedruler (a Neusis construction)
as illustrated above (Courant and Robbins 1996).
An approximate trisection is described by Steinhaus (Wazewski 1945; Peterson 1983; Steinhaus 1999, p. 7). To construct this approximation of an angle having measure , first bisect and then trisect chord (left figure above). The desired approximation is then angle having measure (right figure above). To connect with , applying the law of sines
to triangles and gives
(1)
so .
Since we also have ,
this can be written
(2)
Solving for
then gives
(3)
This approximation is within of even for angles as large as , as illustrated above and summarized in the following
table (Peterson 1983), where angles are measured in degrees.
(OEIS A158599 and A158600), which is readily seen to be a very good approximation to .
Budin (1971) gives another approximate trisection using a compass and straightedge, applicable to positive angles
up to .
Let
be the angle to be trisected, with . Draw the semicircle with
diameter on the side away from , and choose on it with . Choose on the line segment with . Then choose with on the same side of the line as . The circle centered at through and intersects the minor circular
arc
of the circle centered at at . The angle approximates one-third of . For , this construction gives , an error of about .
and 
radians (
and 
, respectively), which can be trisected. Furthermore,
some angles are geometrically trisectable, but cannot be
constructed in the first place, such as 
(Honsberger 1991). In addition, trisection of an arbitrary
angle can be accomplished using a marked ruler (a Neusis construction)
as illustrated above (Courant and Robbins 1996).
having measure 
, first bisect 
and then trisect chord 
(left figure above). The desired approximation is then angle 
having measure 
(right figure above). To connect 
with 
, applying the law of sines
to triangles 
and 
gives
.
Since we also have 
,
this can be written
![sint=2[sin(1/2alpha)cost-sintcos(1/2alpha)].](/images/equations/AngleTrisection/NumberedEquation2.svg)
then gives
of 
even for angles 
as large as 
, as illustrated above and summarized in the following
table (Peterson 1983), where angles are measured in degrees.
(
)
(
)
(
)
(
)
has Maclaurin
series
.
.
Let 
be the angle to be trisected, with 
. Draw the semicircle with
diameter 
on the side away from 
, and choose 
on it with 
. Choose 
on the line segment 
with 
. Then choose 
with 
on the same side of the line 
as 
. The circle centered at 
through 
and 
intersects the minor circular
arc 
of the circle centered at 
at 
. The angle 
approximates one-third of 
. For 
, this construction gives 
, an error of about 
.
