A regular hexagon is one of the simplest polygons to construct exactly with a compass and straightedge. When it is inscribed in a circle, all six vertices lie on the circumference. This construction matters because it shows a direct link between circles, radii, central angles, and equal chord lengths.
It also appears in tiling patterns, engineering layouts, molecular structures, and many design problems.
Key Facts
- A regular hexagon has 6 equal sides and 6 equal interior angles.
- For a regular hexagon inscribed in a circle, side length s = radius r.
- The central angle between adjacent vertices is 360 degrees / 6 = 60 degrees.
- Each side of the hexagon is a chord of the circle.
- Connecting the center to all vertices divides the hexagon into 6 equilateral triangles.
- The perimeter of an inscribed regular hexagon is P = 6r.
Vocabulary
- Regular hexagon
- A six-sided polygon with all sides equal and all interior angles equal.
- Inscribed polygon
- A polygon whose vertices all lie on a circle.
- Radius
- A segment from the center of a circle to any point on the circle.
- Chord
- A segment whose endpoints both lie on a circle.
- Central angle
- An angle with its vertex at the center of a circle and sides passing through points on the circle.
Common Mistakes to Avoid
- Changing the compass width during the construction is wrong because the hexagon depends on stepping the same radius around the circle.
- Placing vertices inside the circle instead of on the circumference is wrong because an inscribed hexagon must have every vertex on the circle.
- Assuming the side is the diameter is wrong because each side of the inscribed regular hexagon equals the radius, not twice the radius.
- Drawing unequal arcs around the circle is wrong because the six arcs must mark off equal 60 degree central angles.
Practice Questions
- 1 A circle has radius 5 cm. What is the side length and perimeter of a regular hexagon inscribed in the circle?
- 2 A regular hexagon inscribed in a circle has perimeter 42 inches. What is the radius of the circle?
- 3 Explain why stepping the compass width equal to the radius around a circle creates exactly six vertices for a regular hexagon.