Heptagon Explained: Definition, Properties, Formulas, and Solved Examples
You just opened your geometry textbook and spotted a shape with seven sides. Your first thought: Is this even a real shape? It is and it shows up in more places than you think, from British coins to famous buildings.
A heptagon (also called a septagon) is a polygon with exactly 7 sides and 7 angles. Whether you are preparing for a Class 7 maths test or trying to understand NCERT geometry concepts at a deeper level, this guide covers everything you need.
By the end of this post, you will know:
- The exact definition and all properties of a heptagon
- How to calculate interior angles, exterior angles, area, and perimeter
- How many diagonals a heptagon has and how to prove it
- Real-life examples you can actually remember
- Five worked problems, from easy to hard
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
Frequently Asked Questions About Heptagon
| Shape | Sides | Interior Angle Sum | Each Angle (regular) | Diagonals |
|---|---|---|---|---|
| Hexagon | 6 | 720° | 120° | 9 |
| Heptagon | 7 | 900° | 128.57° | 14 |
| Octagon | 8 | 1080° | 135° | 20 |
Difference: A hexagon tiles perfectly (like honeycombs) because its 120° angles fit together with no gaps. A heptagon cannot tile a flat surface — its 128.57° angles leave gaps or cause overlaps. An octagon is commonly seen in stop signs.
Real-world heptagons include:
- UK 50 pence coin — a Reuleaux heptagon (curved 7-sided shape) that rolls like a circle, making it detectable by vending machines
- UK 20 pence coin — same Reuleaux design for the same engineering reason (Royal Mint)
- Architectural floor plans — heptagonal rooms and amphitheatres used in some modern buildings
- Specialty tamper-proof bolts — 7-sided cross-section resists standard wrenches
| Regular Heptagon | Irregular Heptagon | |
|---|---|---|
| Sides | All 7 equal | Unequal |
| Angles | All ≈ 128.57° | Different sizes |
| Symmetry | 7 lines of symmetry | None (usually) |
| Angle sum | 900° | Still 900° |
The angle sum is always 900° regardless of whether the heptagon is regular or irregular — this is the most commonly tested distinction.
The perimeter of a regular heptagon is:
Perimeter = 7 × side length (a)
Example: If each side = 9 cm → Perimeter = 7 × 9 = 63 cm
For an irregular heptagon, add all 7 sides individually: P = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇
The area of a regular heptagon with side length a is:
Area ≈ 3.634 × a²
The precise formula is (7/4) × a² × cot(π/7), but 3.634 × a² is the standard approximation used in Class 6–10 exams.
Example: If a = 5 cm → Area ≈ 3.634 × 25 = 90.85 cm²
A heptagon has 14 diagonals. Use the diagonal formula:
Diagonals = n(n − 3) ÷ 2 = 7 × (7 − 3) ÷ 2 = 7 × 4 ÷ 2 = 28 ÷ 2 = 14
A diagonal is a line segment joining two non-adjacent vertices. Each vertex in a heptagon connects to 4 non-adjacent vertices, giving 7 × 4 = 28, divided by 2 (each diagonal shared by 2 vertices) = 14.
Each exterior angle of a regular heptagon is approximately 51.43°, calculated as:
360° ÷ 7 ≈ 51.43°
Quick check:
- Interior angle + Exterior angle = 128.57° + 51.43° = 180°
- The sum of all 7 exterior angles = 7 × 51.43° ≈ 360°
Each interior angle of a regular heptagon is approximately 128.57°. Since all 7 angles are equal in a regular heptagon, divide the total angle sum by 7:
900° ÷ 7 ≈ 128.57°
Note: This is a repeating decimal (128.571428...), so in exams round to 128.57° unless instructed otherwise.
The sum of all interior angles of a heptagon is 900°. You can find this using the formula:
Sum = (n − 2) × 180° = (7 − 2) × 180° = 5 × 180° = 900°
This formula works for any polygon just substitute the number of sides for n.
A heptagon is a flat, closed shape with exactly 7 straight sides and 7 angles. The word comes from Greek hepta means seven and gon means angle. It is also called a septagon. When all 7 sides are equal in length and all 7 angles are equal in size, it is called a regular heptagon.