Ruler and Compass Basics: Essential Constructions for BeginnersRuler-and-compass construction is a foundational part of classical geometry. Using only an unmarked straightedge (ruler) and a compass, you can create a wide range of precise figures: perpendiculars, parallels, angle bisectors, regular polygons, and more. This article explains the basic tools and step-by-step methods for the essential constructions every beginner should learn, why they work, and a few simple exercises to practice.
Tools and conventions
- The straightedge (ruler) can draw a straight line through any two points but cannot measure distances or mark units.
- The compass can draw a circle with any center and radius, and transfer a distance from one location to another.
- Constructions start from given points, lines, or circles. Each step must use only the straightedge and compass operations above.
- We follow Euclidean conventions: lines are infinite, circles are exact, and constructions are exact in the idealized geometric sense.
Fundamental ideas (why these constructions are possible)
Most ruler-and-compass constructions rely on two primitive operations:
- Drawing a line through two known points (straightedge).
- Drawing a circle with a given center and radius equal to the distance between two known points (compass).
Combining those lets you find intersections of lines and circles. Intersections produce new points you can use for subsequent steps. Using symmetry and congruent triangles is the common reasoning to show why constructions produce the desired result.
Essential constructions (step-by-step)
Below are the classical basic constructions with concise step descriptions and brief justifications.
- Constructing a perpendicular bisector of a segment AB
- Place compass at A; draw a circle with radius > half AB.
- With same radius, draw another circle centered at B.
- Let the two circle intersections be P and Q.
- Draw line PQ; it is the perpendicular bisector of AB (it meets AB at its midpoint and is perpendicular by congruent triangles APB and AQB).
- Finding the midpoint of segment AB
- Construct the perpendicular bisector of AB as above; the intersection point M of PQ with AB is the midpoint.
- Constructing a perpendicular from a point C to line l
- If C is off line l: choose two points A and B on l equidistant from the foot you want; draw circles centered at C with radius CA (or CB) to intersect l at two points, etc. Simpler: draw any circle centered at C that meets l at two points P and Q; construct the perpendicular bisector of PQ — it passes through C and meets l at the foot (by symmetry).
- If C is on l: to construct a line through C perpendicular to l, draw a circle centered at C that meets l at two points P and Q; construct the perpendicular bisector of PQ — it goes through C and is perpendicular to l.
- Constructing the angle bisector of angle ∠BAC
- With center A, draw an arc that intersects AB at D and AC at E.
- With centers D and E and the same radius (greater than half DE), draw arcs that intersect at F.
- Draw line AF; it bisects ∠BAC (by congruent triangles formed by the arcs).
- Copying an angle ∠BAC at point D
- Draw an arc centered at A intersecting BA and CA at points E and F.
- With same radius, draw an arc centered at D intersecting the ray where the copied angle will sit at G.
- Measure the distance EF with the compass; with center G draw an arc of radius EF to intersect the previous arc at H.
- Connect D to H; ∠HD? equals ∠BAC.
- Constructing a line parallel to a given line l through point P
- Choose a point A on l. Construct an angle at P equal to angle formed by line AP and l (copy angle method). The new line through P will be parallel to l.
- Constructing an equilateral triangle on segment AB
- Draw circle centered at A with radius AB.
- Draw circle centered at B with radius BA.
- The intersection point(s) C of the circles with line AB form equilateral triangle(s) ABC (sides AB = BC = CA).
- Constructing a regular hexagon inscribed in a circle
- For a circle with center O, any radius equals side length of inscribed regular hexagon. From one point on the circle, step off the radius along the circumference using the compass — repeating six times gives the vertices.
- Constructing a perpendicular through a point on a segment and dividing segments into n equal parts
- For dividing a segment AB into n equal parts: from A draw any ray at an angle; mark off n equal segments along the ray using the compass; connect the last mark to B; draw parallels through the other marks to meet AB; the intersection points divide AB into n equal parts (this uses Thales’ theorem).
- Constructing the circumcenter and incenter of a triangle
- Circumcenter: perpendicular bisectors of any two sides meet at the circumcenter O (center of circumscribed circle).
- Incenter: angle bisectors of any two angles meet at the incenter I (center of inscribed circle).
Common exercises for practice
- Construct perpendicular bisector and midpoint of several segments of different lengths.
- Bisect various angles: acute, obtuse, right.
- Construct the circumcircle and incircle of a triangle.
- Divide a segment into 3, 4, 7 equal parts.
- Inscribe a regular pentagon and hexagon in a given circle (pentagon requires extra steps — see extended methods below).
Short proofs and intuition
- Perpendicular bisector: intersection points of the two equal-radius circles are equidistant from A and B; the line through them is locus of points equidistant from A and B — that’s the perpendicular bisector.
- Angle bisector: points on the angle bisector are equidistant from the sides of the angle. The arc intersection construction creates equal chords and congruent triangles establishing equality of the two sub-angles.
Limitations — “impossible” constructions
Some classic problems are impossible with only ruler and compass: trisecting an arbitrary angle, doubling the cube (constructing cube root of 2), and constructing a general regular 7-gon. These impossibilities stem from algebraic facts about constructible numbers: with ruler and compass you can only construct lengths obtained from the rationals by a finite sequence of additions, subtractions, multiplications, divisions, and square roots. Numbers requiring solutions of irreducible cubic or higher-degree equations with Galois groups not a 2-group are not constructible.
A few tips for neater constructions
- Keep compass width consistent where required (copying distances).
- Make arcs large enough to avoid near-tangent intersections.
- Label intermediate points and follow a consistent order of steps to avoid confusion.
Resources to learn more (books and terms to search)
- Euclid’s Elements (Books I–III for constructions)
- “Geometry: Euclid and Beyond” by Robin Hartshorne
- Classical construction compendia and interactive geometry software (GeoGebra)
Per practice: try constructing the perpendicular bisector and angle bisector for several different figures, then use those to find the circumcenter and incenter of the same triangle.