Geometric construction and plane geometry
This domain tests compass-and-straightedge accuracy and your understanding of why a construction works. The exam expects construction arcs and lines to remain visible, thin and precise, with the finished geometry drawn thick.
Core constructions
- Perpendicular bisector of a line; bisection of an angle; dropping a perpendicular from a point to a line.
- Dividing a line into n equal parts: draw a ray at any convenient angle, step n equal divisions with dividers, join the last division to the end of the line, then draw parallels.
- External tangent from a point P to a circle centre O: bisect OP, draw a semicircle on OP, and the intersection with the circle gives the tangent point.
- Fillet of radius R between two lines: offset both lines by R; their intersection is the centre; tangent points lie on perpendiculars dropped to each line.
- Regular polygons: hexagon by stepping the circumradius around the circle; general polygon method using a semicircle divided into n parts; octagon inside a square using corner arcs.
- Conics and curves: ellipse by concentric circles or the rectangle method; parabola from a focus and directrix; involute by unwinding a divided circle; cycloid by rolling a divided circle along a straight line.
Worked example
Construct a regular hexagon across flats 60 mm. Across corners = 60 / cos 30 degrees = 69.28 mm, so the circumradius is 34.64 mm. Draw a circle of radius 34.6 mm, mark a point on it and step the same radius six times around the circumference. Joining alternate points gives the hexagon and its across-flats check is 60 mm.
Traps
Using the across-flats value as the circle diameter; leaving arcs that overshoot the tangent point so a kink appears; failing to mark tangent points; erasing construction lines that carry marks; blunt compass leads producing double lines.