Lesson 13.1.4

Inscribing Equilateral Triangles & Hexagons

A circle's radius is the same length as the side of an inscribed regular hexagon — making the hexagon the easiest polygon to inscribe. Connect every other vertex for the equilateral triangle.

Introduction

Purple: inscribed hexagon. Green dashed: equilateral triangle.

Past Knowledge

Copying segments (13.1.1). Circle properties (10.1). Central angles.

Today's Goal

Inscribe a regular hexagon and equilateral triangle in a circle.

Future Success

Inscribing squares (13.1.5), tessellations, architectural design.

Key Concepts

The Key Insight

In a regular hexagon inscribed in a circle, each side = the radius. This is because the central angle is 60° → each triangle formed from center is equilateral.

Step-by-Step Constructions

Inscribed Regular Hexagon

  1. 1Draw a circle and mark a starting point on it.
  2. 2Set compass to the radius.
  3. 3“Walk” the compass around the circle, marking each intersection (6 marks total).
  4. 4Connect consecutive marks → regular hexagon.

Inscribed Equilateral Triangle

  1. 1Inscribe a hexagon (steps above).
  2. 2Connect every other vertex (skip one each time).

Why It Works

Central angle of a regular hexagon = 360°/6 = 60°. Each triangle from the center has two sides = radius and an included 60° angle.

Since two sides are equal (radii) with a 60° included angle, the base angles are each 60° → equilateral triangle → side = radius. ✓

For the inscribed equilateral triangle: connecting every other vertex spans 2 × 60° = 120° central angles, giving 3 equal chords → equilateral.

∎ 60° central angles guarantee side = radius for hexagon; 120° arcs guarantee equilateral triangle.

Common Pitfalls

Getting 5 or 7 Marks

If the compass width drifts, you won't land back where you started after 6 steps. Keep the compass locked at exactly the radius.

Real-Life Applications

Honeycomb Patterns

Bees build hexagonal cells because hexagons tessellate perfectly. The construction shows why — the radius naturally creates 6 equal divisions.

Hex Bolt Heads

Hex bolts are regular hexagons inscribed in circles. CNC machines follow the same geometric principle to cut the shape.

Practice Quiz

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