Geometric Constructions: Triangles, Angles and Regular Polygons
A geometric construction draws a figure using only an unmarked straightedge and a compass, and every step is justified by a property of lines and circles. In brief: with these two tools you can copy segments and angles, build triangles from three sides (SSS), two sides and the included angle (SAS) or two angles and a side (ASA), and construct regular polygons such as the equilateral triangle, the square, the regular hexagon and the regular pentagon.
Constructions were the heart of Greek geometry, and they train precise thinking: you must know why each step works. This guide lists the basic constructions, shows how to build triangles from given data, how to construct regular polygons with a compass and straightedge, and which regular polygons can be constructed this way.
What are the rules of compass and straightedge construction?
The straightedge has no markings and draws a line through two points. The compass draws a circle with a given center through a given point, and in the classical rules it does not carry a length from place to place, although this can be done with a construction. Allowed steps: draw a line through two points, draw a circle with a given center and radius, and mark the intersection points of lines and circles.
Measuring with a ruler or a protractor is not allowed in a pure construction. The advantage is exactness: for example, a construction of a 60° angle is exact, while a measured angle can be a little off.
What are the basic constructions?
- Copy a segment: draw a ray, set the compass to the segment length and mark it on the ray.
- Perpendicular bisector of a segment AB: draw two circles of equal radius (more than half of AB) centered at A and B. The line through their two intersection points is the perpendicular bisector and also gives the midpoint.
- Angle bisector: draw an arc centered at the vertex that cuts both sides; from the two cut points draw arcs of equal radius that meet inside the angle; the ray through the vertex and that meeting point bisects the angle.
- Copy an angle: draw the same arc on both angles and transfer the chord length of the arc.
- Perpendicular from a point to a line: draw an arc centered at the point that cuts the line twice, then construct the perpendicular bisector of the segment between the cut points.
- Parallel line through a point: copy the angle (corresponding angles) or construct a rhombus.
How do you construct a triangle from given data?
The three congruence criteria give three standard constructions.
SSS: three sides a, b, c
- Draw a segment BC with length a.
- Draw a circle with center B and radius c, and a circle with center C and radius b.
- Let A be an intersection point of the circles. Triangle ABC has the required sides.
The circles meet only if the triangle inequality holds: the sum of any two sides must be greater than the third. For 3, 4 and 8, the circles do not meet since 3 + 4 < 8.
SAS: two sides and the included angle
Draw one side, construct the given angle at one end and mark the second side on the arm of the angle. Join the endpoints.
ASA: two angles and the included side
Draw the side, construct the given angles at its ends and let the arms meet. The two arms meet only if the sum of the two angles is less than 180°.
Special triangles
An equilateral triangle with side a: draw circles with center A and center B and radius a. Their intersection is the third vertex. A right triangle with given hypotenuse c: construct the midpoint M of the hypotenuse, draw the circle with center M and radius c/2, and any point on this circle (other than the ends) is a right-angle vertex, by Thales’ theorem.
How do you construct regular polygons?
Equilateral triangle and hexagon
The radius of the circumscribed circle of a regular hexagon equals its side. So draw a circle and, with the same radius, step around the circle from a point on it: six marks divide the circle into six arcs of 60°. Connect them to get the hexagon, and connect every second mark to get an equilateral triangle.
Square
Draw a circle and a diameter. Construct the perpendicular diameter. The four endpoints are the vertices of a square. Bisecting the angles gives the regular octagon, and repeated bisection gives the 16-gon and so on.
Regular pentagon
One construction uses the golden ratio. Draw a circle with center O, a diameter and the perpendicular radius OB. Find the midpoint M of the radius to the right of O. With center M and radius MB, draw an arc to the diameter at P. The distance BP is the side of the regular pentagon inscribed in the circle. Mark the side length BP around the circle five times.
Constructible regular polygons
| Sides n | Constructible? | Central angle |
|---|---|---|
| 3 | Yes | 120° |
| 4 | Yes | 90° |
| 5 | Yes | 72° |
| 6 | Yes | 60° |
| 7 | No | about 51.43° |
| 8 | Yes | 45° |
| 9 | No | 40° |
| 10 | Yes | 36° |
| 11 | No | about 32.73° |
| 12 | Yes | 30° |
| 13 | No | about 27.69° |
| 14 | No | about 25.71° |
| 15 | Yes | 24° |
| 16 | Yes | 22.5° |
| 17 | Yes | about 21.18° |
| 18 | No | 20° |
| 19 | No | about 18.95° |
| 20 | Yes | 18° |
Gauss proved in 1796 that the regular 17-gon is constructible. In 1837 Pierre Wantzel showed that the Gauss criterion is also necessary: a regular n-gon is constructible exactly when n is a power of 2 times a product of distinct Fermat primes (3, 5, 17, 257, 65537).
Where do constructions come from?
Euclid’s Elements starts with constructions such as the equilateral triangle on a given segment. Three classical problems, doubling the cube, trisecting a general angle and squaring the circle, occupied mathematicians for two thousand years. They were proved impossible with compass and straightedge in the nineteenth century, using algebra, which shows how algebra can answer geometric questions.
Where are constructions used today?
Technical drawing and architecture still use compass-and-straightedge ideas. Dynamic geometry software, such as GeoGebra, reproduces them on screen. Constructions also teach logical reasoning: each step is a statement that must be justified, which is useful preparation for proofs.
Which mistakes are common?
Changing the compass opening
Keep the opening when the construction requires equal radii, such as the bisector arcs.
Using a ruler to measure
A construction is exact only if the straightedge is used for lines alone.
Arcs that are too small
Intersection arcs must be large enough to cross. For the perpendicular bisector, the radius must exceed half of the segment.
Erasing construction marks
Keep the arcs visible. They show that the construction is valid.