Polygons: Properties, Construction, and Angle Sums

By the end of this lesson you will be able to:

• State what makes a polygon regular
• Construct an equilateral triangle, square, and regular hexagon
• Find the interior angle sum of any polygon using
• Find each exterior angle of a regular polygon using

Which Shapes Work for Tiling the Floor?

A tiler wants to cover a floor with one shape — no gaps, no overlaps.

• Squares? Triangles? Hexagons?
• How does she know every angle without a protractor?

Recall from construction-02:

• How did we build an exact angle?
• How did we build an exact angle?

Four Shapes on the Tiler's Sample Board

"Which of these four shapes look the same all around — every side equal, every angle equal?"

A Polygon: Closed Figure, Straight Sides Only

• Vertex — corner where two sides meet
• Interior angle — angle inside the corner
• Exterior angle — angle between side and extension of next side

Regular Needs Both Conditions to Hold

Check-In: Regular or Irregular? Give a Reason.

1. Is an equilateral triangle regular?

2. Is a scalene triangle regular?

For each: answer yes/no and state which conditions are met.

All Three Constructions Reuse Old Primitives

Every move today recycles a technique from construction-02:

• Equal arcs → → equilateral triangle
• Perpendicular → → square
• Unchanged compass radius → regular hexagon

No new technique is introduced — only old moves in new combinations.

Constructing an Equilateral Triangle: 60° Arc

Given: cm

1. Draw cm
2. Open compass to cm; arc from
3. Same opening; arc from
4. Arcs cross at ; join and

Constructing a Square: Base and Perpendiculars

Given: cm

1. Draw cm
2. Raise at → perpendicular
3. Raise at → perpendicular

Two perpendiculars, both exactly — this guarantees the right angles of the square.

Constructing a Square: Complete the Shape

Continuing from the previous step

4. Mark on and on , each cm up
5. Join

Constructing a Regular Hexagon: Step the Radius

Circle radius 3 cm — do not change the compass opening

1. Mark start point on circle
2. Step the radius six times around
3. Sixth step lands on start — accuracy check
4. Join the six marks (radius = hexagon side: that is why it closes)

Check-In: Match Each Construction to Its Primitive

Options: 60° arc · 90° perpendicular · unchanged compass opening

Bridge: Does Every Triangle Sum to 180°?

Equilateral triangle: .

But is only true because it is equilateral?

We need the triangle sum to hold for all triangles to triangulate any polygon.

Proof That Every Triangle Sums to 180°

• and reappear at the apex as alternate angles
• , , lie on a straight line →

Uses the alternate-angle result from construction-01.

Interior Angle Sum Formula for Any Polygon

Split from one vertex into triangles:

• Quadrilateral → triangles →
• Pentagon → triangles →
• -gon → triangles

For a regular -gon: each interior

Caution: Dividing by n Requires Regularity

gives the sum for any polygon.

Dividing by works only if regular.

Irregular quadrilateral:

• Sum ✓ (formula holds for any polygon)
• Each angle ≠ ✗ (not all equal)

÷ n needs regularity.

Probe: Spot the Error in This Student's Work

A student writes:

"Each interior angle of a regular hexagon "

1. What did the student calculate correctly?
2. What error did they make?
3. What is the correct answer?

Applying the Interior Formula to a Regular Hexagon

:

Formula and construction agree: at every vertex — this is the tiler's self-check.

Bridge: The Exterior Angle as a Turn

We know every interior angle. But a tiler walks the outside edge.

At each corner she turns to face the next side.

"What does she turn through at each corner, and what is the total for one full circuit?"

The exterior angle is the turn at each vertex.

Exterior Angle Sum Is Always 360°

• Turn through the exterior angle at each vertex
• One full circuit = one full turn

Each Exterior Angle of a Regular Polygon

For a regular -gon: all exterior angles are equal.

Side and extension form a straight line, so:

Two Routes Confirm the Same Hexagon Angles

Route 1 (exterior → interior):

Route 2 (interior formula):

Both give interior, exterior. ✓ Use disagreement as an error signal.

Practice: Complete the Angle-Sum Table

Last column is your accuracy check.

Exit Task: Regular Octagon — Show All Working

A regular octagon has sides.

(a) Interior angle sum

(b) Each interior angle

(c) Each exterior angle

(d) Verify interior + exterior

Key Results: The Tiler Now Knows Every Angle

• Interior sum: — any polygon
• Each interior (regular): — only if regular
• Each exterior (regular):
• At every vertex: interior exterior

Next: Construction-04 — scale drawings where every angle is exact.