# Pentagon Shape - Properties, Area, and Perimeter

## TL;DR
A pentagon is a polygon with 5 sides and 5 interior angles summing to 540°. A regular pentagon has all sides equal and all angles equal to 108° each. Its area formula is A=\(\frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2\approx 1.72 s^2\), and its perimeter is P=5s.

## What Is a Pentagon?
A **pentagon** is a polygon with **5 sides** and **5 vertices**. The name comes from Greek _penta_ (five) + _gonia_ (angle).

**Sum of interior angles** of any pentagon:
\((n−2)×180°=(5−2)×180°=540°\)

This is true for _any_ pentagon — convex or concave, regular or irregular.

## Types of Pentagons
### Regular Pentagon
All 5 sides equal length; all 5 angles equal (each 108°). The most symmetric pentagon. Five axes of symmetry; rotational symmetry of order 5.

### Irregular Pentagon
Sides and/or angles unequal. Most pentagons in real-world geometry problems are irregular.

### Convex Pentagon
All interior angles < 180°. All diagonals lie inside the pentagon. Most common type.

### Concave Pentagon
At least one interior angle > 180° (a "dent"). At least one diagonal exits the pentagon.

## Properties of a Regular Pentagon
- **All sides equal**: each of length s.
- **All interior angles equal**: each 108°.
- **All exterior angles equal**: each 72°.
- **Sum of interior angles**: 540°.
- **5 axes of symmetry** — each through one vertex and the midpoint of the opposite side.
- **5-fold rotational symmetry** — rotates onto itself every 72°.
- **Diagonals form a pentagram** (a 5-pointed star) — when all 5 diagonals are drawn.
- **Diagonal-to-side ratio** is the **golden ratio** \(\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618\).

## Area Formulas
### Regular Pentagon — Exact Formula
\[ A=\frac{1}{4}\sqrt{5(5+2\sqrt{5})} \cdot s^2 \text{ or } A\approx 1.72s^2 \]

### Regular Pentagon — Using Apothem
The **apothem** \(a\) is the distance from the centre to the midpoint of a side.
\[ A=\frac{1}{2} \cdot P \cdot a = \frac{5sa}{2} \]

### Irregular Pentagon
No single formula. Divide into triangles, compute each triangle's area, sum them.

## Perimeter of a Pentagon
For _any_ pentagon:
\[ P=s_1+s_2+s_3+s_4+s_5 \]

For a **regular** pentagon with side s:
\[ P=5s \]

## Three Worked Examples — Quick, Standard, Stretch
### Quick — Perimeter
A regular pentagon has side 7 cm. Find its perimeter.
\[ P=5×7=35 \text{ cm} \]

### Standard — Area Using Side
A regular pentagon has side 6 m. Find its area.
\[ A\approx 1.72×6^2\approx 61.94 \text{ m}^2 \]

### Stretch — Area Using Apothem
A regular pentagon has side 4 cm and apothem 2.75 cm. Find its area.
\[ A=\frac{5\cdot4\cdot2.75}{2}=27.5 \text{ cm}^2 \]

## Why Does the Pentagon Matter? (The Real-World GROUND)
Pentagons appear in nature, architecture, and science:
- **The Pentagon (US Department of Defense headquarters)** is a regular-pentagon-shaped building.
- **Sea stars (starfish)** have 5-fold radial symmetry.
- **Flowers** — many flower species have 5 petals.
- **Pentaprisms** — used in cameras and surveying instruments.
- **Crystalline structures** — pentagonal symmetry found in quasicrystals.
- **Soccer balls** — have pentagonal panels.

## A Worked Example
Find the interior angle of a regular pentagon.
**The correct method.** 
Sum of interior angles: \( (n−2)×180°=(3×180°)=540° \)
For a regular pentagon, each interior angle =\( 540°/5=108° \).

## What Are the Most Common Mistakes With Pentagons?
### **Mistake 1: Using triangle area formulas directly**
**The fix:** Use the pentagon formula or divide into triangles for irregular ones.

### **Mistake 2: Confusing interior and exterior angles**
**The fix:** Interior angle = 108°. Exterior angle = 72°.

### **Mistake 3: Forgetting that the formula assumes "regular"**
**The fix:** The area formula is for a _regular_ pentagon.

## Key Takeaways
- **A pentagon** has 5 sides, 5 vertices, and interior angles summing to 540°.
- **A regular pentagon** has all sides equal and all angles =108°.
- **Area of a regular pentagon**: A≈1.72s².
- **Perimeter of a regular pentagon**: P=5s.
- **The golden ratio** \(\varphi\) appears throughout the regular pentagon.