Heptagon Explained: Definition, Properties, Formulas, and Solved Examples
What is a Heptagon?
A heptagon is a closed 2D polygon with 7 sides, 7 vertices, and 7 angles. From Greek hepta (seven) + gonia (angle). Also known as a septagon. When all 7 sides and angles are equal, it is a regular heptagon.
Properties of a Heptagon
| Property | Value |
|---|---|
| Number of sides | 7 |
| Number of vertices | 7 |
| Sum of interior angles | 900° |
| Each interior angle (regular) | ≈ 128.57° |
| Each exterior angle (regular) | ≈ 51.43° |
| Number of diagonals | 14 |
| Lines of symmetry (regular) | 7 |
| Rotational symmetry order | 7 |
Types of Heptagon
| Type | Sides Equal? | Angles Equal? | All Angles < 180°? |
|---|---|---|---|
| Regular | Yes | Yes | Yes |
| Irregular | No | No | Usually |
| Convex | No | No | Yes |
| Concave | No | No | No (one > 180°) |
Interior and Exterior Angles
Sum of interior angles:
Sum = (n − 2) × 180° = (7 − 2) × 180 = 5 × 180 = 900°
Each interior angle (regular):
900° ÷ 7 = ≈ 128.57°
Each exterior angle (regular) — step-by-step:
- Sum of all exterior angles of any polygon = 360°
- All exterior angles in a regular heptagon are equal
- Each = 360° ÷ 7 = ≈ 51.43°
- Check: 128.57° + 51.43° = 180° ✓
Heptagon Formulas
Perimeter (regular) = 7 × aArea (regular) ≈ 3.634 × a²
Where a = side length
Area derivation: A regular heptagon divides into 7 identical isosceles triangles meeting at the centre. Summing their areas using the apothem gives the approximation 3.634 × a².
Number of Diagonals
Formula: n(n − 3) ÷ 2
- Substitute: 7 × (7 − 3) ÷ 2
- Simplify: 7 × 4 ÷ 2
- Calculate: 28 ÷ 2 = 14 diagonals
Heptagon in Real Life
- UK 50p coin — Reuleaux heptagon; constant diameter allows vending machine detection (Royal Mint)
- UK 20p coin — same Reuleaux design for the same engineering reason
- Architecture — heptagonal floor plans used in amphitheatres and conference rooms
- Specialty bolts — 7-sided cross-section resists standard wrenches (tamper resistance)
Heptagon vs Other Polygons
| Property | Pentagon | Hexagon | Heptagon | Octagon |
|---|---|---|---|---|
| Sides | 5 | 6 | 7 | 8 |
| Interior angle sum | 540° | 720° | 900° | 1080° |
| Each interior angle | 108° | 120° | 128.57° | 135° |
| Each exterior angle | 72° | 60° | 51.43° | 45° |
| Diagonals | 5 | 9 | 14 | 20 |
| Lines of symmetry | 5 | 6 | 7 | 8 |
Pattern: each additional side adds 180° to the angle sum and more diagonals.
Solved Examples
Ex 1 (Easy): Each side = 6 cm. Perimeter? → 7 × 6 = 42 cm
Ex 2 (Easy): Sum of interior angles? → (7−2) × 180 = 900°
Ex 3 (Medium): Six angles are 120°, 130°, 140°, 125°, 135°, 110°. Find the 7th. → 900° − 760° = 140°
Ex 4 (Medium): Area with side = 8 cm? → 3.634 × 64 = ≈ 232.58 cm²
Ex 5 (Hard): In regular heptagon ABCDEFG, triangle ABC — find angle CAB. → Interior angle at B = 128.57° → angles at A and C = (180° − 128.57°) ÷ 2 = 25.71° each ✓
Common Mistakes to Avoid
- Using n = 6 instead of 7 → getting 720° instead of 900°
- Rounding 51.43° to 51° too early — causes angle sum check to fail
- Forgetting to divide by 2 in the diagonal formula → getting 28 instead of 14
- Applying regular heptagon angle (128.57°) to irregular heptagon problems
Quick Revision Summary
- 7 sides · 7 vertices · 7 angles
- Interior angle sum = 900°
- Each interior angle (regular) ≈ 128.57°
- Each exterior angle (regular) ≈ 51.43°
- Perimeter = 7a | Area ≈ 3.634a²
- Diagonals = 14 | Symmetry lines = 7
- Real life: UK 50p & 20p coins, architecture, tamper-proof bolts