Inscribed Polygons in a Circle

Lesson 1 of 1: Hexagon, Triangle, and Square

In this lesson:

• Construct a regular hexagon, equilateral triangle, and square inscribed in a circle
• Explain why each construction works using circle properties

Learning Objectives for Today's Lesson

1. Construct an equilateral triangle inscribed in a given circle
2. Construct a square inscribed in a given circle
3. Construct a regular hexagon inscribed in a given circle
4. Explain why each construction works geometrically
5. Identify relationships among all three inscribed polygons

You Already Know the Tools

You've used compass and straightedge to:

• Copy a segment and copy an angle
• Bisect a segment and bisect an angle
• Construct perpendicular and parallel lines

Today's goal: use those same tools to build exact regular polygons inside a circle.

What Does the Word "Inscribed" Mean?

A polygon is inscribed in a circle when every vertex lies on the circle.

• The circle is called the circumscribed circle (circumcircle)
• For a regular polygon: vertices are equally spaced around the circle
• Equal spacing → equal arcs → equal sides → equal angles

Inscribed Versus Not Inscribed: Key Difference

Every vertex on the circle — or construction fails.

The Key Fact: Hexagon Side Equals Radius

Triangle has :

• , are radii → equal
• drawn with compass at → also
• All sides equal → equilateral →

Quick Check: Why Does It Work?

Triangle has all three sides equal to .

Why does that force ?

Think about it before the next slide…

Hexagon Construction Steps One Through Three

• Step 1: Draw circle with center , radius
• Step 2: Mark any point on the circle
• Step 3: Set compass to — do not change it — mark arc from to get

Hexagon Construction: Walk and Connect Vertices

• Step 4–6: Move compass to , mark ; to , mark ; to , mark ; to , mark
• Step 7: Connect ––––––

The sixth arc lands exactly back on — the geometry guarantees it.

Why the Hexagon Closes Perfectly

Six equilateral triangles fill the circle exactly:

• Each , , … has at
• — full rotation around
• All sides equal → equal arcs → equal sides

Regular by construction, not by measurement.

From Hexagon: Connect Every Other Vertex

Connect , , — the alternating hexagon vertices.

Why Triangle Is Equilateral

Each side spans two hexagon arcs:

• All three arcs = → equal
• Equal arcs → equal chords →
• Equal sides → equilateral → each angle

Triangle Side Length:

In triangle : ,

Quick Check: Triangle Side Length

The hexagon side = .

What is the equilateral triangle side length?

Try to recall before advancing…

Answer: Equilateral Triangle Side Length

Each triangle side spans two hexagon edges — the 30-60-90 relationship explains why:

• The hexagon diagonal (across two edges) creates a 30-60-90 triangle
• Long leg = , consistent with the law of cosines result

The Square Needs a Different Strategy

For the hexagon and triangle, the compass "walked" around the circle.

For a square: 4 equally spaced points at 90° each — use perpendicular diameters.

• A diameter passes through the center
• Two perpendicular diameters create four arcs
• Their four endpoints are the square's vertices

Watch Out: Diameters, Not Just Chords

The chords must pass through the center to make a square.

Square Construction: All Four Steps

1. Draw circle, mark center
2. Draw diameter through
3. Perpendicular bisector of → hits circle at ,
4. Connect ––––

Why the Square Construction Works

Perpendicular diameters create four equal central angles:

• Equal central angles → equal arcs → equal chords
• Each square angle = inscribed angle in semicircle =

Regular by construction.

Square Side Length Derived from Pythagorean Theorem

Right triangle : , :

Quick Check: Square Side vs. Diagonal

A circle has radius .

A student says the inscribed square's side = 10 (the diameter). What is the error?

Identify the mistake before the next slide.

Answer: Diagonal Equals Diameter, Side Does Not

The diagonal of the inscribed square = (the diameter).

The side = — shorter than the diameter.

The side connects adjacent vertices; the diagonal connects opposite vertices through the center.

All Three Polygons: Same Circle

Three regular polygons — one circle — three different compass strategies.

Comparing the Three Inscribed Polygons

Perimeters of Inscribed Polygons Approach Circumference

Circumference = — all three perimeters fall below it.

Quick Check: Comparing Side Lengths

The three polygons are all inscribed in a circle of radius .

Rank the side lengths from shortest to longest.

Think before advancing…

Answer: Ranking Side Lengths Shortest to Longest

More sides → shorter individual side, but more sides total → larger perimeter.

Key Takeaways and Misconception Warnings

✓ Hexagon: compass walks at radius — equilateral

✓ Triangle: connect alternating vertices , ,

✓ Square: perpendicular diameters through

Keep compass at radius — never adjust

Square: diameters through center, not just chords

Side ; diagonal

What Comes Next: Circle Theorems

HSG.C.A builds directly on today:

• Inscribed angle theorem: angles formed by chords at the circle
• Arc-chord relationships: equal arcs equal chords
• Circumscribed circles: the circumcircle you used today

Today's constructions are the foundation for circle theory.