Perpendicular bisector · Angle bisector · Regular hexagon · Richmond's regular pentagon — animated step by step
A compass-and-straightedge construction is a figure built using only two idealized tools, in the tradition set out by Euclid around 300 BCE: a collapsing compass (it can draw a circle of a given radius centred at a given point, but does not "remember" a radius once lifted from the page — though it can be shown that a collapsing compass and a modern fixed-radius compass are equally powerful) and an unmarked straightedge (it draws the infinite line through two points but carries no measuring marks). Everything you see animated here — the perpendicular bisector, the angle bisector, the regular hexagon and the regular pentagon — is built from exactly these two moves, repeated: draw a circle, draw a line, and mark the points where existing circles and lines cross. Every intersection point on this page is computed with the real circle–circle and line–circle intersection formulas, not hard-coded coordinates, so the constructions are geometrically exact.
Not everything is constructible this way, and this is one of the most celebrated results in mathematics. The three classical problems of antiquity — doubling the cube (constructing a cube with twice the volume of a given cube), trisecting an arbitrary angle, and squaring the circle (constructing a square with the same area as a given circle) — all resisted construction for over two thousand years, and all three were eventually proven impossible. Pierre Wantzel proved in 1837, using field-extension degree arguments, that doubling the cube and trisecting a general angle with compass and straightedge alone cannot be done. Squaring the circle was proven impossible later still, in 1882, when Ferdinand von Lindemann showed that π is a transcendental number (not the root of any polynomial with rational coefficients), which rules out constructing a length of √π from a unit length.
The underlying reason is algebraic: every point you can reach with compass and straightedge, starting from two given points, has coordinates lying in a tower of quadratic extensions of the rational numbers — each new construction step at worst takes a square root of a previously constructible quantity, because intersecting lines and circles only ever involves solving linear or quadratic equations. A length is therefore constructible only if it can be reached from the rationals by a finite chain of steps, each of which adjoins at most a square root. Trisecting a 60° angle in general would require constructing cos(20°), which is a root of the irreducible cubic 4x³ − 3x − 1 = 0 — a degree-3 extension of the rationals, which can never sit inside a tower built purely from square roots (whose degrees are always powers of 2). That is why no amount of cleverness with an unmarked straightedge and collapsing compass can trisect a general angle exactly.
The four constructions animated on this page — bisecting a segment perpendicularly, bisecting an angle, inscribing a regular hexagon, and inscribing a regular pentagon by Richmond's method — are, by contrast, all genuinely constructible, because a regular n-gon is constructible exactly when n factors as a power of 2 times a product of distinct Fermat primes (3, 5, 17, 257, 65537 …); both 6 = 2×3 and 5 satisfy that condition, which is why compass-and-straightedge hexagons and pentagons have been known since antiquity.
Carl Friedrich Gauss, at age 19 in 1796, proved that the regular 17-gon (heptadecagon) is constructible with compass and straightedge — the first progress on the problem since antiquity — and was reportedly so proud of it that he asked for a regular 17-gon to be engraved on his tombstone (the stonemason declined, saying it would look too much like a circle). The complete answer, that a regular n-gon is constructible exactly when n is a power of 2 times a product of distinct Fermat primes, was finished by Pierre Wantzel in 1837 — the same paper in which he proved angle trisection and cube-doubling impossible. Only five Fermat primes are currently known (3, 5, 17, 257, 65537), so it remains unknown whether infinitely many constructible regular polygons with odd numbers of sides exist.
This simulator animates four classical compass-and-straightedge constructions on a 2D canvas, computing every circle, line and intersection with real geometric formulas — nothing is pre-baked. It also explains, honestly, which classical problems (doubling the cube, trisecting a general angle, squaring the circle) were proven impossible with these tools, and why the hexagon and pentagon shown here are not among them.
Four constructions: the perpendicular bisector of a segment, the bisector of an angle, a regular hexagon inscribed in a circle, and a regular pentagon built with Richmond's method. Each is broken into discrete compass/straightedge moves, and every new circle, line and intersection point is computed with the standard circle–circle and line–circle intersection formulas.
Choose a construction from the dropdown, then either drag the Step slider or press Play to watch it unfold move by move. The current step is drawn in bright green while earlier steps fade to a muted tone. Toggle step labels on or off, and use Reset to start over.
Gauss proved the regular 17-gon constructible in 1796 at age 19. Wantzel completed the theory of constructible polygons — and separately proved angle trisection and cube-doubling impossible — in a single 1837 paper.
Only two moves are allowed: drawing the circle centred at one already-constructed point and passing through another, and drawing the infinite line through two already-constructed points. New points are only created where such circles and lines cross. The straightedge has no marks and the compass is not used to transfer a measured distance except via an actual drawn circle.
No. Pierre Wantzel proved in 1837 that trisecting an arbitrary angle with an unmarked straightedge and compass alone is impossible in general. Trisecting 60°, for instance, would require constructing cos(20°), a root of the irreducible cubic 4x³ − 3x − 1 = 0, and constructible numbers can only ever come from towers of square-root (degree-2) extensions of the rationals, never a genuine cube root of that kind. Some special angles (like 90°) can be trisected, but a general method does not exist.
Yes. Constructing a square with the same area as a given circle of radius 1 requires constructing a segment of length √π. Ferdinand von Lindemann proved in 1882 that π is transcendental — it is not the root of any polynomial with rational coefficients — and every compass-and-straightedge-constructible number must be algebraic (in fact confined to towers of square-root extensions), so √π cannot be constructed.
Gauss and Wantzel showed a regular n-gon is constructible exactly when n is a power of 2 multiplied by any number of distinct Fermat primes (3, 5, 17, 257, 65537). Since 6 = 2 × 3 and 5 itself is a Fermat prime, both the regular hexagon and the regular pentagon satisfy this condition, which is why clean classical constructions for them, like the one by Richmond used here for the pentagon, have existed since antiquity.
Every circle–circle intersection uses the standard formula based on the distance between centres and the two radii (solving for the radical line and the perpendicular offset), and every line–circle intersection is found by substituting the line's parametric equation into the circle's equation and solving the resulting quadratic. No coordinates are hard-coded; they are recalculated live from the construction's defining points and radii.