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Construct equilateral triangle, square, regular hexagon inscribed in circle

20 practice questions 2 video lessons Theory + worked examples

Constructing Regular Polygons

California Geometry • Standard G-CO.13 • Geometric Constructions

Constructing Regular Polygons is a topic in Geometric Constructions in the California Common Core State Standards. It is aligned to Standard G-CO.13, which requires students to construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Inscribing regular polygons constructs an equilateral triangle, a square, and a regular hexagon inside a circle with compass and straightedge.

California Geometry › Geometric Constructions › Constructing Regular Polygons  —  Standard G-CO.13

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Practice questions

Every question with a fully worked solution.

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Watch 2 video(s)
  • Geometric Constructions: Inscribe a Regular Hexagon, Triangle and Square in a Circle Watch
  • Constructing an equilateral triangle inside a given circle Watch
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Theory

A regular polygon inscribed in a circle has all vertices on the circle, equally spaced. The central angle between consecutive vertices is \(\dfrac{360^\circ}{n}\).

  • Regular hexagon: the side equals the radius, so step the compass (set to the radius) six times around the circle.
  • Equilateral triangle: connect every other hexagon vertex, or place three points \(120^\circ\) apart.
  • Square: draw two perpendicular diameters; their endpoints are the vertices.
The hexagon is the easiest: its side length is exactly the radius, a direct consequence of six \(60^\circ\) equilateral triangles.
Regular hexagon inscribed in a circle Stepping the radius around a circle six times marks the vertices of a regular hexagon. r hexagon: side = radius (6 arc steps)
A regular hexagon: the radius steps around the circle six times.
Inscribed equilateral triangle and square Equally spaced points on a circle give an inscribed equilateral triangle (every 120 degrees) or square (every 90 degrees). equilateral (120° apart) & square (90°)
An inscribed equilateral triangle and square.

The central angle:

\[\text{central angle}=\dfrac{360^\circ}{n};\qquad \text{hexagon side}=\text{radius}\]
the central angle is 360 over n; a regular hexagon's side equals the radius
Hexagon = 6 equilateral triangles around the center, each with a \(60^\circ\) central angle.

How to inscribe a regular hexagon

  1. Set the compass to the circle's radius.
  2. Mark a starting point on the circle.
  3. Step the compass around the circle, marking six points.
  4. Connect the points in order.
Example 1 — Hexagon side
How does the side of an inscribed regular hexagon relate to the circle's radius?
Solution

The side equals the radius, so stepping the radius around the circle six times marks the vertices.

\(\text{side}\)\(=\)\(\text{radius}\)
the hexagon side equals the radius
Example 2 — Central angle of a hexagon
What is the central angle between consecutive vertices of a regular hexagon?
Solution

Divide the full turn by 6.

\(\dfrac{360^\circ}{6}\)\(=\)\(60^\circ\)
the central angle is 60 degrees
Example 3 — Equilateral triangle
At what central angle apart are the vertices of an inscribed equilateral triangle?
Solution

Three equally spaced points: \(\dfrac{360^\circ}{3}=120^\circ\) apart.

\(\dfrac{360^\circ}{3}\)\(=\)\(120^\circ\)
120 degrees apart
Example 4 — Square
How is a square inscribed in a circle constructed?
Solution

Draw two perpendicular diameters; their four endpoints on the circle are the square's vertices (every \(90^\circ\)).

draw two perpendicular diameters for the four vertices

Common pitfalls

Hexagon side = radius, exactly. This is unique to the regular hexagon.
Keep the compass width fixed at the radius while stepping around.
A square uses perpendicular diameters, not the radius-stepping method.

Frequently asked questions

How do you inscribe a regular hexagon in a circle?

Set the compass to the radius and step it around the circle six times; connect the six marks.

Why does the hexagon's side equal the radius?

Because a regular hexagon is made of six equilateral triangles meeting at the center, each with sides equal to the radius.

What is the central angle of an inscribed regular polygon?

\(\dfrac{360^\circ}{n}\) between consecutive vertices, where \(n\) is the number of sides.

How do you inscribe a square in a circle?

Draw two perpendicular diameters; their four endpoints on the circle are the square's vertices.

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