Triangles, squares, and regular hexagons

Polygons are the most basic flat shapes, and many of them can be constructed exactly with a ruler and compass. We'll learn how to build an equilateral triangle, a square, and a regular hexagon , three of the cleanest constructions in elementary geometry.

Concept

Equilateral triangle

A triangle with all three sides equal , and (because of this) all three angles equal to $60°$.

Construction. Given a segment $\overline{AB}$ as the base:

1. Place the compass tip at $A$, with radius $AB$, and draw an arc above the segment.
2. Without changing the radius, place the tip at $B$ and draw another arc that crosses the first.
3. Call the intersection $C$. Draw $\overline{AC}$ and $\overline{BC}$.
4. Triangle $ABC$ is equilateral, because $AB = AC = BC$ (all equal to the compass setting).

This is the very first proposition of Euclid's Elements. Two arcs, three sides , done.

Square

A four-sided polygon with all sides equal and all angles $90°$.

Construction. Given a segment $\overline{AB}$ as the base:

1. Construct a perpendicular to $\overline{AB}$ at $A$ (call it line $\ell_A$).
2. Construct a perpendicular to $\overline{AB}$ at $B$ (call it line $\ell_B$).
3. On $\ell_A$, mark a point $D$ such that $AD = AB$ (use the compass to copy the length).
4. On $\ell_B$, mark a point $C$ such that $BC = AB$ (on the same side as $D$).
5. Connect $\overline{CD}$. The quadrilateral $ABCD$ is a square.

Regular hexagon

A six-sided polygon with all sides equal and all angles $120°$. The hexagon's beautiful secret: the side length of a regular hexagon equals the radius of its circumscribed circle.

Construction.

1. Draw a circle of any radius $r$ centred at $O$. This will be the circumscribed circle of the hexagon.
2. Choose any point $A$ on the circle.
3. Without changing the compass spread (still $r$), place the tip at $A$ and mark a new point on the circle. Call it $B$.
4. From $B$, mark another point at distance $r$ on the circle: $C$.
5. Continue: $D$, $E$, $F$. The sixth mark will land back at $A$.
6. Connect $\overline{AB}, \overline{BC}, \dots, \overline{FA}$.

The result is a perfect regular hexagon $ABCDEF$ inscribed in the circle.

This works because each radius $\overline{OA}, \overline{OB}, \dots$ together with two consecutive vertices forms an equilateral triangle (since $OA = OB = AB = r$). Six such triangles fit around the centre, each contributing a $60°$ angle at $O$, totalling $360°$.

Why some regular polygons are easy, and others not

• Triangle (3), square (4), hexagon (6), octagon (8 , by bisecting a square), 12-gon (by bisecting a hexagon), 16-gon, etc. , all constructible.
• The regular pentagon (5 sides) is also constructible, but harder. Gauss as a teenager famously found a way to construct the regular 17-gon!
• The regular heptagon (7 sides), 9-gon, 11-gon, etc. are not constructible with ruler and compass alone , a deep fact proved in the 19th century.

Worked examples

Example 1. Construct an equilateral triangle with side $5$ cm.

• Draw $\overline{AB}$, $5$ cm.
• Compass radius $5$ cm. Arc from $A$, arc from $B$. Intersection: $C$.
• Connect $A, B, C$.

Example 2. Construct a square with side $4$ cm.

• Draw $\overline{AB}$, $4$ cm.
• Construct perpendiculars at $A$ and $B$ (using the perpendicular construction from earlier).
• Mark $C$ and $D$ at distance $4$ cm along the perpendiculars (same side).
• Connect $\overline{CD}$.

Example 3. Construct a regular hexagon of side $3$ cm.

• Draw a circle of radius $3$ cm centred at $O$.
• Pick a point $A$ on the circle.
• With compass spread $3$ cm, mark $A \to B \to C \to D \to E \to F \to A$.
• Connect consecutive vertices.

Example 4. Verify that the hexagon you constructed has equal sides.

• Each side was drawn with the same compass spread ($r$). So all six sides are equal.

Try it yourself

1. Construct an equilateral triangle with side $6$ cm.
2. Construct a square with side $5$ cm.
3. Construct a regular hexagon with side $4$ cm.
4. Inside your hexagon, draw the six radii from the centre to each vertex. You should get six equilateral triangles. Verify by measuring.
5. Inside your hexagon, draw the three long diagonals (from $A$ to $D$, $B$ to $E$, $C$ to $F$). What is their common point?
6. Construct an equilateral triangle, then bisect each of its angles to find its centre.
7. Construct a $30$°-$60$°-$90$° triangle starting from an equilateral triangle (bisect one angle).
8. Challenge: construct a regular dodecagon ($12$-gon) by starting with a regular hexagon and bisecting each of its central angles.

Activity

Hexagon flower. Construct a regular hexagon. At each of its vertices, place a small circle of radius equal to half the hexagon's side. You should see a six-petalled flower forming. Try shading alternate petals , you'll get a beautiful symmetric design that has been used in Indian rangoli and Islamic tile art for centuries.