Regular Polygons

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 ($120°$).

Regular or Not?

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.

All Three Angles of a Regular Octagon

Find the interior angle, the exterior angle and the central angle of a regular octagon.

Count the sides: An octagon has $n = 8$ sides = $n = 8$

Interior angle: $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8}$ = $135°$

Exterior angle: $\frac{360°}{8}$ (check: $180° - 135° = 45°$) = $45°$

Central angle: $\frac{360°}{8}$, the full turn around the center shared by 8 triangles = $45°$

Answer: Interior angle $135°$, exterior angle $45°$, central angle $45°$.

Finding n from the Interior Angle

Each interior angle of a regular polygon is $144°$. How many sides does it have?

Find the exterior angle: $180° - 144°$ = $36°$

Use exterior angle $= \frac{360°}{n}$: $n = \frac{360°}{36°}$ = $n = 10$

Check with the interior angle formula: $\frac{(10-2) \times 180°}{10} = \frac{1440°}{10} = 144°$ = Correct

Answer: The polygon has 10 sides: it is a regular decagon.

Area with a Given Apothem

A regular pentagon has sides of 8 cm and an apothem of 5.5 cm. Find its area.

Find the perimeter: $P = 5 \times 8$ = $P = 40$ cm

Write the area formula: $A = \frac{1}{2} \cdot P \cdot a$ = Formula ready

Substitute: $A = \frac{1}{2} \cdot 40 \cdot 5.5$ = $A = 110$ cm²

Answer: The area of the pentagon is $110$ cm².

Area of a Regular Hexagon from Its Side

A regular hexagon has sides of 6 cm. Find its area without being given the apothem.

Split it into 6 triangles from the center: Central angle $\frac{360°}{6} = 60°$, base angles $\frac{180° - 60°}{2} = 60°$ = 6 equilateral triangles with side 6 cm

The apothem is the height of one triangle: $a = \sqrt{6^2 - 3^2} = \sqrt{27}$ = $a \approx 5.196$ cm

Area of one triangle: $\frac{1}{2} \cdot 6 \cdot 5.196$ = $\approx 15.59$ cm²

Area of the hexagon: $6 \times 15.588 \approx 93.53$ (check: $\frac{1}{2} \cdot 36 \cdot 5.196 \approx 93.53$) = $A \approx 93.53$ cm²

Answer: The area is about $93.53$ cm² (exactly $54\sqrt{3}$ cm²).

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.

Mistake: Calling a rhombus regular because all its sides are equal

Why: 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.

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

Why: $(n-2) \times 180°$ 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 $\frac{720°}{6} = 120°$.

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

Why: 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 $n = \frac{360°}{36°} = 10$.

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

Why: 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.

Honeycombs and Hexagon Tiles

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

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

Stop Signs

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

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

Building a Gazebo Floor

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

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

A regular polygon has all sides equal AND all angles equal; a rhombus and a rectangle are not regular

Interior angle of a regular $n$-gon: $\frac{(n-2) \times 180°}{n}$

Exterior angle and central angle are both $\frac{360°}{n}$

To find $n$ from an interior angle, first take the exterior angle $180° - \text{interior}$, then $n = \frac{360°}{\text{exterior}}$

Perimeter $P = n \cdot s$ and area $A = \frac{1}{2} \cdot P \cdot a$, where the apothem $a$ goes from the center to the midpoint of a side

Only the equilateral triangle, the square and the regular hexagon tile a floor on their own, because $60°$, $90°$ and $120°$ divide $360°$

Q: Is every triangle with three equal sides regular?

A: 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.

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

A: 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 $\frac{360°}{n}$.

Q: Why can a regular pentagon not tile a floor?

A: 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.

Q: How many lines of symmetry does a regular polygon have?

A: 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.

Exit Presentation

Regular Polygons

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Geometry / Polygons

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 min

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