A regular polygon has all sides equal and all interior angles equal. The compass-and-ruler constructions for a regular hexagon and an equilateral triangle rely on one elegant fact: the radius of the circumscribed circle equals the side length of both shapes, so you can step out the vertices with the compass set to the radius.
What equipment do you need and why?
Compass — draws arcs and circles at a consistent radius. Ruler (straight edge) — draws straight line segments. Pencil — faint construction lines are required; ink lines are harder to correct.
The restriction to compass and ruler (no protractor, no set square) is classical: it means you are using only exact geometric relationships, not estimated angles. All the constructions on this page leave visible construction arcs, which examiners expect to see.
How do you construct an equilateral triangle?
An equilateral triangle has three equal sides and three 60° angles.
Method — given a line segment AB of length s:
- Set the compass to length s (the distance AB).
- Place the compass point on A. Draw an arc above the line segment.
- Without changing the compass width, place the compass point on B. Draw a second arc that crosses the first arc. Label the crossing point C.
- Draw line segments AC and BC.
Triangle ABC is equilateral: AC = BC = AB = s (all arcs drawn with the same radius).
Why it works: point C is equidistant from both A and B by the same distance s, so all three sides are equal. All interior angles are therefore 60°.
How do you construct a regular hexagon inside a circle?
A regular hexagon has six equal sides. When inscribed in a circle of radius r, every side also has length r — so the compass set to r steps around the circle exactly six times.
Method:
- Draw a circle of radius r and mark the centre O.
- Mark any point P on the circumference as your starting vertex.
- Set the compass to radius r (do not change this setting throughout).
- Place the compass point on P and mark the next point around the circle.
- Move the compass point to the new mark and repeat. Continue until you have six equally spaced points on the circumference.
- Connect the six points in order with straight line segments.
The result is a regular hexagon with side length r.
Verification: six equilateral triangles, each with sides r, meet at the centre O. Central angle per triangle = 360° ÷ 6 = 60°, matching the angles of an equilateral triangle.
How do you construct a regular hexagon on a line segment?
If you need the hexagon to have a specific side length s (given as a drawn segment):
- Construct the first equilateral triangle on the base segment AB (method above). Label the apex C.
- The centre of the hexagon is the midpoint M of AB.
Actually, this becomes easier using the inscribed-circle method: take s as the circle's radius.
The cleanest KS3 approach is: draw the circle with radius equal to the required side length, then use the stepping method above.
What is the interior angle of a regular hexagon?
Interior angle = (n − 2) × 180° ÷ n, where n = 6:
Interior angle = 4 × 180° ÷ 6 = 720° ÷ 6 = 120°
Each vertex of the hexagon has a 120° angle. The exterior angle is 180° − 120° = 60°.
How do other regular polygons compare?
| Polygon | Sides (n) | Interior angle | Constructable with compass and ruler? |
|---|---|---|---|
| Equilateral triangle | 3 | 60° | Yes |
| Square | 4 | 90° | Yes |
| Regular pentagon | 5 | 108° | Yes (more complex) |
| Regular hexagon | 6 | 120° | Yes |
| Regular octagon | 8 | 135° | Yes |
| Regular heptagon | 7 | ~128.6° | No — not constructable exactly |
The equilateral triangle and hexagon are the most commonly tested KS3 constructions because they depend only on the simple radius-equals-side property.
What mistakes should you avoid?
Mistake 1 — Changing the compass width during the hexagon construction. The compass must stay at radius r throughout all six steps. Any change produces unequal sides.
Mistake 2 — Pressing the compass point too hard. Damaging the paper shifts the radius. Use a firm but gentle press.
Mistake 3 — Rubbing out construction arcs. The arcs are part of your working. Examiners will not award full marks for a construction without visible arcs.
Frequently asked questions
Why does stepping the radius around a circle give exactly six points?
The circumference of a circle is 2πr. Each chord of length r subtends an angle of 60° at the centre (because the triangle formed by the two radii and the chord is equilateral). Six × 60° = 360°, which completes the full circle. So six steps of length r divide the circle into exactly six equal parts.
Can you construct a regular pentagon with a compass and ruler?
Yes, though the method is more complex than for a hexagon. It involves constructing a golden ratio, which requires additional bisection steps. A regular pentagon is sometimes tested at GCSE Higher but rarely at KS3.
How do you construct a square using compass and ruler?
Draw a line segment AB. At A, construct a perpendicular line (using the standard perpendicular construction). Mark point D on this perpendicular at distance AB from A. With the compass set to AB, mark point C from both B (above) and D (to the right). Draw the four sides. A square has four equal sides and four 90° angles.
Is a regular hexagon made up of equilateral triangles?
Yes — a regular hexagon can be divided into six equilateral triangles, all meeting at the centre. This is why the construction works: each step around the circle creates one of these triangles. It also makes the hexagon useful in tiling (tessellation), since equilateral triangles and regular hexagons are among the only regular polygons that tile a flat surface without gaps.
For KS3 geometry coaching that explains the reasoning behind every construction, visit aitutors.me.