Polygons • Topic 3 of 3

Regular Polygons and Tessellations

What is a Regular Polygon? A regular polygon is a polygon that is both equiangular (all angles equal) and equilateral (all sides equal).

Properties of Regular Polygons:

All sides have the same length
All interior angles have the same measure
All exterior angles have the same measure
Regular polygons are always convex
They have rotational symmetry and reflection symmetry

What is a Tessellation? A tessellation (or tiling) is a repeating pattern of shapes that covers a plane completely with no gaps and no overlaps.

Which Regular Polygons Tessellate? Only three regular polygons tessellate by themselves:

Regular Polygon	Interior Angle	Fits around point?	Tessellates?
Equilateral triangle	60°	6 × 60° = 360°	Yes
Square	90°	4 × 90° = 360°	Yes
Regular hexagon	120°	3 × 120° = 360°	Yes

Why others don't tessellate:

Pentagon (108°): no whole number × 108° = 360°
Octagon (135°): no whole number × 135° = 360°
Heptagon (≈128.6°): no whole number × 128.6° = 360°

Semi-regular Tessellations: Combinations of two or more regular polygons that tessellate together (e.g., octagons and squares).

Solved Examples

Worked Example

Find the measure of each exterior angle of a regular decagon.

Solution- Decagon has $n = 10$ sides - Each exterior angle = $\frac{360°}{n} = \frac{360°}{10} = 36°$ - **Answer:** 36° *Example 2: Can a regular pentagon tessellate by itself? Explain. Solution: - Regular pentagon interior angle = $\frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°$ - To tessellate, multiple copies must fit around a point: $k \times 108° = 360°$ - $k = 360° \div 108° = 3.333...$ (not an integer) - Therefore, regular pentagons cannot tessellate by themselves - **Answer:** No, because 3.33 pentagons would be needed to surround a point *Example 3: A regular polygon has each interior angle = 150°. How many sides does it have? Solution: - Interior angle = $\frac{(n-2) \times 180°}{n} = 150°$ - Multiply both sides by n: $(n-2) \times 180° = 150° \times n$ - $180n - 360 = 150n$ - $30n = 360$ - $n = 12$ - **Answer:** 12 sides (dodecagon)

Key Points

A regular polygon has equal sides and equal angles
Only 3 regular polygons tessellate alone: equilateral triangle, square, regular hexagon
For a shape to tessellate, the interior angle must divide 360° evenly
Tessellations cover a plane with no gaps and no overlaps
Semi-regular tessellations use two or more types of regular polygons
Many natural patterns (honeycombs, tiles) use tessellations
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A regular polygon has equal sides and equal:

Explanation: Equal angles.

Q2.A tessellation covers a plane with:

Explanation: No gaps or overlaps.

Q3.Which does NOT tessellate?

Explanation: $108^\circ$ does not divide $360^\circ$.

Q4.At a tessellation vertex, the angles sum to:

Explanation: $360^\circ$.

Practice more MCQs → 📝 Assignment · 35 marks