Teacher Guide: Regular Polygons

Learn what makes a polygon regular, how to find its interior, exterior and central angles, its perimeter and area, and which regular polygons tile a floor.

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10 questions on Polygons. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Decide whether a polygon is regular by checking both sides and angles
Calculate the interior, exterior and central angles of a regular polygon
Find the number of sides of a regular polygon from an interior or exterior angle
Calculate the perimeter and the area of a regular polygon using the apothem
Explain why only triangles, squares and hexagons tile the plane as regular polygons

Prerequisites

• Names of polygons by number of sides (Introduction to Polygons, Classifying Polygons)
• The interior angle sum $(n-2) \times 180°$ (Polygon Angle Sum Formula)
• Exterior angles and their sum of $360°$ (Exterior Angles of Polygons)
• Area of a triangle $\frac{1}{2} \cdot b \cdot h$ and the Pythagorean theorem

Discussion Starters

1. Is a square the only quadrilateral that is regular? How would you convince a friend?
2. Why do you think bees build hexagons and not squares or triangles?
3. What happens to the interior angle when a regular polygon gets more and more sides? Can it ever reach $180°$ ?
4. Why is it easier to find the number of sides from the exterior angle than from the interior angle?

Common Misconceptions

Equal sides are enough for a polygon to be regular

Remediation: Push a square made of 4 equal sticks into a rhombus: the sides stay equal, the angles change. Then show a rectangle: equal angles, unequal sides. Regular needs both.

The apothem and the radius are the same length

Remediation: Draw both in a hexagon: the radius ends at a vertex, the apothem at the midpoint of a side. The apothem is a leg and the radius the hypotenuse of a right triangle, so the radius is always longer.

Any regular polygon can tile a floor if you arrange it cleverly

Remediation: Let students try with paper pentagons. Then compute $\frac{360°}{108°} \approx 3.33$ : the angles around a point must add to exactly $360°$ , so the number must be whole.

Differentiation Ideas

For Struggling Students:

• Give a table with n, angle sum and interior angle already started for n = 3 to 6
• Use paper polygons and a protractor to measure angles before using formulas
• Give the apothem in every area problem

For On-Level Students:

• Find n from interior and exterior angles
• Calculate areas of regular polygons with given apothems
• Explain with angles which regular polygons tile the plane

For Advanced Students:

• Find the apothem of a hexagon or a square from the side alone
• Explore the octagon and square floor pattern: why does $135° + 135° + 90° = 360°$ work?
• Show that the area of a regular n-gon gets close to the area of a circle as n grows

Standards Alignment

7.G.B.5 (CCSS.MATH.CONTENT.7.G.B.5)
Use facts about supplementary and adjacent angles to write and solve equations for an unknown angle
7.G.B.6 (CCSS.MATH.CONTENT.7.G.B.6)
Solve real-world problems involving area of two-dimensional objects composed of triangles and polygons
HSG.CO.A.3 (CCSS.MATH.CONTENT.HSG.CO.A.3)
Describe the rotations and reflections that carry a regular polygon onto itself

Lesson Resources

visualRegular Polygon Explorer
Slide the number of sides and watch the central, interior and exterior angles, the apothem, the perimeter and the area change live
activityTiling Test
Students try to tile a sheet with paper triangles, squares, pentagons, hexagons and octagons and record which ones leave gaps
worksheetRegular Polygon Table
Fill a table for n = 3 to 12 with interior, exterior and central angles and look for patterns

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A regular polygon has all sides equal AND all angles equal. Both conditions are needed:

A rhombus has four equal sides, but its angles are not all equal (for example $60°$ and $120°$ ), so it is not regular.
A rectangle has four equal angles ( $90°$ ), but its sides are not all equal, so it is not regular.
A square has both, so it is a regular quadrilateral.

Every regular polygon has a center

O

, the same distance from every vertex. Lines from the center to the vertices cut a regular

n

-gon into

n

congruent isosceles triangles.

**Angles of a regular

n

-gon** (the angle sum

(n-2) \times 180°

comes from the lesson on the angle sum formula):

\text{interior angle} = \frac{(n-2) \times 180°}{n} \qquad \text{exterior angle} = \frac{360°}{n} \qquad \text{central angle} = \frac{360°}{n}

The central angle is the angle at the center between lines to two neighboring vertices. For a regular polygon it equals the exterior angle.

Lengths: the radius

r

goes from the center to a vertex, the apothem

a

goes from the center to the midpoint of a side and is perpendicular to it.

Perimeter and area (side

s

P = n \cdot s \qquad A = \frac{1}{2} \cdot P \cdot a

Tiling: a regular polygon tiles a floor on its own only if its interior angle divides

360°

. That works for exactly three: the equilateral triangle (

60°

), the square (

90°

) and the regular hexagon (

120°

Worked Examples

Decide which shapes are regular polygons: a rhombus with angles of

70°

and

110°

, a rectangle measuring 3 cm by 5 cm, and an equilateral triangle.

Check the rhombus

All 4 sides are equal, but the angles are

70°

and

110°

→ Angles not equal: not regular

Check the rectangle

All 4 angles are

90°

, but the sides are 3 cm and 5 cm → Sides not equal: not regular

Check the equilateral triangle

All 3 sides are equal and all 3 angles are

60°

→ Regular

Answer: Only the equilateral triangle is regular. The rhombus fails on angles, the rectangle fails on sides.

Common Mistakes

Calling a rhombus regular because all its sides are equal

Why it's wrong: Equal sides are only half of the definition. A rhombus can have angles of $60°$ and $120°$ .

Correct: A polygon is regular only if all sides AND all angles are equal. Check both every time.

Forgetting to divide by $n$ : giving $(n-2) \times 180°$ as one interior angle

Why it's wrong: $(n-2) \times 180°$ is the SUM of all interior angles, for a hexagon $720°$ , which is far too big for one angle.

Correct: One interior angle of a regular $n$ -gon is $\frac{(n-2) \times 180°}{n}$ , for a hexagon $\frac{720°}{6} = 120°$ .

Finding $n$ by dividing $360°$ by the INTERIOR angle

Why it's wrong: For an interior angle of $144°$ this gives $\frac{360}{144} = 2.5$ sides, which is impossible. Only the exterior angle fits $\frac{360°}{n}$ .

Correct: First find the exterior angle $180° - 144° = 36°$ , then $n = \frac{360°}{36°} = 10$ .

Using the side or the radius instead of the apothem in $A = \frac{1}{2} P a$

Why it's wrong: The radius goes to a vertex and is longer than the apothem. Only the apothem meets the side at a right angle, so only it is the height of each triangle.

Correct: Use the apothem: the perpendicular distance from the center to the midpoint of a side.

Why It Matters

Regular polygons are the shapes people choose when everything must be fair, strong or easy to repeat:

Nature: Bees build hexagonal honeycomb cells. Hexagons fill the space with no gaps and use the least wax for the walls.
Traffic signs: The stop sign is a regular octagon, so drivers recognize its shape even from behind.
Floors and walls: Tiles are squares, triangles or hexagons, the only regular polygons that cover a floor with no gaps.
Engineering: Nuts and bolt heads are regular hexagons, so a wrench fits them from six directions.
Design and building: Gazebos, tables and windows use regular octagons and pentagons; the central angle tells the carpenter how to cut each piece.

With the formulas in this lesson you can find any angle, the perimeter and the area of a regular polygon from just the number of sides and one length.

Real World Applications

Honeycombs and Hexagon Tiles

Bees and tile makers both use regular hexagons, because three of them fit exactly around one point with no gap.

Example:

Squares fit 4 around a point ( $4 \times 90° = 360°$ ), equilateral triangles fit 6 ( $6 \times 60° = 360°$ ).

1Try It Yourself

A tile maker wants to cover a bathroom floor with regular hexagons only.

How many hexagons meet at each corner point?

Step 1: Write the mathematical expression

Divide the full angle around a point by one interior angle:

Stop Signs

A stop sign is a regular octagon. Sign makers use the area to know how much reflective film each sign needs.

Example:

A regular octagon with side 25 cm has an apothem of about 30.2 cm.

2Try It Yourself

A stop sign is a regular octagon with sides of 25 cm and an apothem of 30.2 cm.

How many square centimeters of reflective film cover the whole sign?

Step 1: Write the mathematical expression

Use $A = \frac{1}{2} \cdot P \cdot a$ :

Building a Gazebo Floor

The floor of an octagonal gazebo is made of 8 equal triangular wooden sections that meet at the center.

Example:

Each section has the central angle $\frac{360°}{8} = 45°$ at its tip at the center.

3Try It Yourself

A carpenter cuts the 8 isosceles triangles for a regular octagonal gazebo floor.

At what angle must the carpenter cut each of the two base corners of one triangle?

Step 1: Write the mathematical expression

Subtract the central angle from $180°$ and share the rest equally:

Key Takeaways

1A regular polygon has all sides equal AND all angles equal; a rhombus and a rectangle are not regular
2Interior angle of a regular $n$ -gon: $\frac{(n-2) \times 180°}{n}$
3Exterior angle and central angle are both $\frac{360°}{n}$
4To find $n$ from an interior angle, first take the exterior angle $180° - \text{interior}$ , then $n = \frac{360°}{\text{exterior}}$
5Perimeter $P = n \cdot s$ and area $A = \frac{1}{2} \cdot P \cdot a$ , where the apothem $a$ goes from the center to the midpoint of a side
6Only the equilateral triangle, the square and the regular hexagon tile a floor on their own, because $60°$ , $90°$ and $120°$ divide $360°$

Frequently Asked Questions

Is every triangle with three equal sides regular?

Yes. For triangles, equal sides force equal angles, so every equilateral triangle has three

60°

angles. From four sides on, this is no longer true: a rhombus has equal sides but can have unequal angles.

Why are the exterior angle and the central angle the same?

Both share the full

360°

equally among

n

parts: the exterior angles are the

n

turns of one walk around the polygon, the central angles are the

n

slices around the center. So both are

\frac{360°}{n}

Why can a regular pentagon not tile a floor?

Its interior angle is

108°

. Three of them make

324°

, leaving a gap of

36°

, and four would make

432°

, which overlaps. Since

\frac{360}{108}

is not a whole number, pentagons cannot meet without gaps.

How many lines of symmetry does a regular polygon have?

A regular

n

-gon has

n

lines of symmetry, for example 6 for a regular hexagon. The lesson on symmetry in polygons looks at these lines and at rotations in detail.

Glossary

Regular polygon: A polygon with all sides equal and all angles equal.
Center: The point inside a regular polygon that is the same distance from every vertex.
Central angle: The angle at the center between the lines to two neighboring vertices: $\frac{360°}{n}$ .
Radius: The distance from the center of a regular polygon to a vertex.
Apothem: The perpendicular distance from the center of a regular polygon to the midpoint of a side.
Tessellation (tiling): A pattern of shapes that covers a flat surface with no gaps and no overlaps.

Formula Card

Interior angle

\frac{(n-2) \times 180°}{n}

One angle inside a regular polygon with $n$ sides

Exterior and central angle

\frac{360°}{n}

The turn at each vertex, and the angle at the center for one side

Number of sides

n = \frac{360°}{\text{exterior angle}}

Exterior angle $= 180° - $ interior angle

Perimeter

P = n \cdot s

Number of sides times the side length

Area

A = \frac{1}{2} \cdot P \cdot a

Half the perimeter times the apothem $a$

Teacher Guide: Regular Polygons

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Honeycombs and Hexagon Tiles

Stop Signs

Building a Gazebo Floor

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Interior angle

Exterior and central angle

Number of sides

Perimeter

Area

More in This Topic

Introduction to Polygons

Diagonals of Polygons

Classifying Polygons

Polygon Angle Sum Formula