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.

Intermediate25 min18 problems
Practice Problems

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°60° and 120°120°), so it is not regular.
  • A rectangle has four equal angles (90°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 OO, the same distance from every vertex. Lines from the center to the vertices cut a regular nn-gon into nn congruent isosceles triangles.
**Angles of a regular nn-gon** (the angle sum (n−2)×180°(n-2) \times 180° comes from the lesson on the angle sum formula):
interior angle=(n−2)×180°nexterior angle=360°ncentral angle=360°n\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 rr goes from the center to a vertex, the apothem aa goes from the center to the midpoint of a side and is perpendicular to it.
Perimeter and area (side ss):
P=n⋅sA=12⋅P⋅aP = 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°360°. That works for exactly three: the equilateral triangle (60°60°), the square (90°90°) and the regular hexagon (120°120°).

Try it now

What is each exterior angle of a regular decagon (10 sides), in degrees?

Worked Examples

Decide which shapes are regular polygons: a rhombus with angles of 70°70° and 110°110°, a rectangle measuring 3 cm by 5 cm, and an equilateral triangle.
1

Check the rhombus

All 4 sides are equal, but the angles are 70°70° and 110°110° → Angles not equal: not regular
2

Check the rectangle

All 4 angles are 90°90°, but the sides are 3 cm and 5 cm → Sides not equal: not regular
3

Check the equilateral triangle

All 3 sides are equal and all 3 angles are 60°60° → Regular

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°60° and 120°120°.

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

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

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

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

Finding nn by dividing 360°360° by the INTERIOR angle

Why it's wrong: For an interior angle of 144°144° this gives 360144=2.5\frac{360}{144} = 2.5 sides, which is impossible. Only the exterior angle fits 360°n\frac{360°}{n}.

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

Using the side or the radius instead of the apothem in A=12PaA = \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.

Interactive Visual

Regular polygon

60°as = 6 cm

Central angle

360∘÷6=60∘360^\circ \div 6 = 60^\circ

Each interior angle

(6−2)⋅180∘6=120∘\dfrac{(6 - 2) \cdot 180^\circ}{6} = 120^\circ

Each exterior angle

180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ

Perimeter

P=n⋅s=6⋅6=36 cmP = n \cdot s = 6 \cdot 6 = 36\,\text{cm}

Apothem (centre to the midpoint of a side)

a=s2tan⁡(180∘÷6)≈5.2 cma = \dfrac{s}{2\tan(180^\circ \div 6)} \approx 5.2\,\text{cm}

Area

A=12⋅P⋅a≈12⋅36⋅5.196≈93.53 cm2A = \tfrac{1}{2} \cdot P \cdot a \approx \tfrac{1}{2} \cdot 36 \cdot 5.196 \approx 93.53\,\text{cm}^2

The radii cut a regular polygon into n congruent isosceles triangles. Each has base s and height a, so the area is n · ½ · s · a = ½ · P · a.

Function graph

-4-2024-4-2024xy3 × 120° = 360°120°120°120°

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which of these shapes is a regular polygon?

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 (4×90°=360°4 \times 90° = 360°), equilateral triangles fit 6 (6×60°=360°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=12⋅P⋅aA = \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 360°8=45°\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°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 nn-gon: (n−2)×180°n\frac{(n-2) \times 180°}{n}
  • 3Exterior angle and central angle are both 360°n\frac{360°}{n}
  • 4To find nn from an interior angle, first take the exterior angle 180°−interior180° - \text{interior}, then n=360°exteriorn = \frac{360°}{\text{exterior}}
  • 5Perimeter P=n⋅sP = n \cdot s and area A=12⋅P⋅aA = \frac{1}{2} \cdot P \cdot a, where the apothem aa 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°60°, 90°90° and 120°120° divide 360°360°

Frequently Asked Questions

Yes. For triangles, equal sides force equal angles, so every equilateral triangle has three 60°60° angles. From four sides on, this is no longer true: a rhombus has equal sides but can have unequal angles.
Yes. For triangles, equal sides force equal angles, so every equilateral triangle has three 60°60° angles. From four sides on, this is no longer true: a rhombus has equal sides but can have unequal angles.
Both share the full 360°360° equally among nn parts: the exterior angles are the nn turns of one walk around the polygon, the central angles are the nn slices around the center. So both are 360°n\frac{360°}{n}.
Its interior angle is 108°108°. Three of them make 324°324°, leaving a gap of 36°36°, and four would make 432°432°, which overlaps. Since 360108\frac{360}{108} is not a whole number, pentagons cannot meet without gaps.
A regular nn-gon has nn 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: 360°n\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

(n−2)×180°n\frac{(n-2) \times 180°}{n}

One angle inside a regular polygon with $n$ sides

Exterior and central angle

360°n\frac{360°}{n}

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

Number of sides

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

Exterior angle $= 180° - $ interior angle

Perimeter

P=n⋅sP = n \cdot s

Number of sides times the side length

Area

A=12⋅P⋅aA = \frac{1}{2} \cdot P \cdot a

Half the perimeter times the apothem $a$

Today's challengeOne problem, three tries

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Diagonals of Polygons

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