Here are a list of elementary constructions that can be carried out with a straightedge and compass. They are arranged roughly in order of difficulty, and if you can do most of these, you can do most standard geometric constructions.

- Copy a segment. In other words, mark off a segment that
exactly matches the length of a given segment on a different
straight line.
- Copy an angle. Given an angle, make another angle of exactly
the same size somewhere else.
- Bisect a segment. This problem was already solved in
Section 1
- Bisect an angle. Given an angle, find a line through the
vertex that divides it in half.
- Construct a line perpendicular to a given line through a point
on the given line.
- Construct a line perpendicular to a given line and passing
through a point not on the given line.
- Given a line and a point not on , construct a new
line that passes through and is parallel to .
- Construct an angle whose size is the sum or difference of
two given angles.
- Given three segments, construct a triangle whose sides have the
same lengths as the segments.
- Construct the perpendicular bisector of a line segment.
- Given three points, construct the circle that passes through
all of them.
- Given a circle, find its center.
- Given a triangle , construct the inscribed and circumscribed
circles. The inscribed circle is a circle that fits inside the
triangle and touches all three edges; the circumscribed circle is
outside the triangle except that it touches all three of the vertices
of the triangle.
- Construct angles of , , ,
, and if you want a challenge, .
- Construct a regular pentagon. (A regular pentagon is a five
sided figure all of whose sides and angles are equal.)
- Given a point on a circle , construct a line through
and tangent to .
- Given a circle and a point not on , construct a line
through and tangent to .
- Given two circles and , find lines internally and
externally tangent to both.