# Heptagon: 7-Sided Polygon, Angles, and Examples

## TL;DR
A heptagon is a seven-sided polygon with 7 sides, 7 vertices, and 7 interior angles that add up to 900°. In a regular heptagon, every side and angle is equal, so each interior angle measures about 128.57° and each exterior angle about 51.43°.

## What Is A Heptagon?
A **heptagon** is a closed, flat shape (a [polygon](/content/math/terms/polygon/index.html)) made of **seven straight sides**. The name comes from the Greek _hepta_ (seven) and _gon_ (angle), so "heptagon" literally means "seven angles." You may also see it written as a **septagon** - an older Latin-rooted name for the same 7-sided shape - or simply a **7-gon**.

Every heptagon, no matter how it is drawn, has exactly **7 sides, 7 vertices** (corner points), and **7 interior angles**. What changes from one heptagon to another is whether those sides and angles are all equal.

### Regular versus irregular, convex versus concave
Heptagons split into pairs of types, and telling them apart is where most exam questions live.

- **Regular heptagon:** all 7 sides equal and all 7 angles equal. This is the "textbook" heptagon with the 128.57° angles.
- **Irregular heptagon:** sides and/or angles differ. It still has 7 sides, but they are not all the same length.
- **Convex heptagon:** every interior angle is less than 180°, so no vertex "caves in."
- **Concave heptagon:** at least one interior angle is greater than 180°, so it has a dent.

A regular heptagon is always convex, but an irregular heptagon can be either convex or concave.

### The angle formulas, and where they come from
The interior angles of any polygon follow one rule, derived by cutting the shape into triangles from a single vertex. A heptagon splits into (7−2)=5 triangles, and each triangle's angles sum to 180°.

Sum of interior angles=(n−2)×180°

For a heptagon, n=7:

(7−2)×180°=5×180°=900°

For a **regular** heptagon, share that total equally across the 7 angles:

Each interior angle=900°/7≈128.57°

The exterior angles of _any_ polygon always sum to 360°, so each exterior angle of a regular heptagon is

360°/7≈51.43°

and each interior and exterior pair adds to 180° (a straight line), which you can check: 128.57°+51.43°=180°.

| Property                           | Regular heptagon |
|-----------------------------------|------------------|
| Number of sides                   | 7                |
| Sum of interior angles            | 900°             |
| Each interior angle               | ≈128.57°        |
| Each exterior angle               | ≈51.43°         |
| Number of diagonals               | 14               |
| Lines of symmetry                 | 7                |

The number of diagonals uses n(n−3)/2=7×4/2=14, and the full [interior angles](/content/math/geometry/interior-angles/index.html) rule generalizes this to any polygon.

## Examples Of Heptagon
These build from applying the angle formula up to reasoning about tiling. One of them shows a wrong turn worth walking through.

### Example 1
**What is the sum of the interior angles of a heptagon?**
Use the polygon rule with n=7:

(n−2)×180°=(7−2)×180°

=5×180°=900°

The interior angles of any heptagon - regular or irregular - always sum to 900°.

### Example 2
**Find each interior angle of a regular heptagon. A tempting shortcut goes wrong first.**
The tempting shortcut is to divide 360° by 7, the way you might for angles "around a point": 360°÷7≈51.43°.

That value is wrong for an _interior_ angle, and you can see why: an interior angle of a polygon this size should be obtuse (well over 90°), not a thin 51.43°. What the shortcut actually found is the _exterior_ angle.

The correct method divides the interior-angle _sum_ by 7:

Each interior angle=900°/7≈128.57°.

The 51.43° from the shortcut is exactly the exterior angle, and indeed 128.57°+51.43°=180°. So the shortcut wasn't nonsense, it just answered a different question.

### Example 3
**An irregular heptagon has six interior angles measuring 130°, 140°, 120°, 150°, 100°, and 110°. Find the seventh angle.**
All seven interior angles must sum to 900°.

130°+140°+120°+150°+100°+110°=750°

Seventh angle=900°−750°=150°.

### Example 4
**How many diagonals does a heptagon have?**
Use the diagonal formula, where n is the number of sides.

n(n−3)/2=7(7−3)/2=7×4/2=14.

A heptagon has 14 diagonals - the line segments joining non-adjacent vertices.

### Example 5
**What is each exterior angle of a regular heptagon?**
The exterior angles of any polygon sum to 360°. For a regular heptagon, divide equally:

Each exterior angle=360°/7≈51.43°.

### Example 6
**Can regular heptagons tile a flat floor with no gaps?**
For shapes to tile a flat plane meeting at a vertex, their angles must add to exactly 360° there.

Each interior angle of a regular heptagon is ≈128.57°. Try fitting them around one point:

2×128.57°=257.14° (leaves a gap)

3×128.57°=385.71° (overlaps, exceeds 360°)

No whole number of 128.57° angles lands on 360°, so **regular heptagons cannot tessellate** a flat floor - a real reason you never see seven-sided floor tiles.

## Why The Heptagon Matters - "Seven Sides That Refuse To Tile"
The heptagon is the first "awkward" polygon a student meets, and its awkwardness is exactly what makes it useful to study.

- **It cannot be drawn with compass and straightedge alone.** Unlike the equilateral triangle, square, pentagon, and hexagon, the regular heptagon is _not_ constructible with only those two classical tools - a fact the ancient Greeks suspected and Gauss's work later helped explain. This is why heptagons feel rarer than pentagons or hexagons.
- **It refuses to tessellate.** Because 360° is not a whole-number multiple of 128.57°, heptagons leave gaps, so honeycomb-style tiling uses hexagons instead. That "gap" is the same reason bees build hexagonal cells, not seven-sided ones.
- **Where the maths is going.** The (n−2)×180° rule you used here scales to every polygon - the [octagon](/content/math/geometry/octagon/index.html), the [decagon](/content/math/geometry/decagon/index.html), and beyond - and it is the same rule that fixes the angles of the [quadrilaterals](/content/math/geometry/quadrilaterals/index.html) you already know. Master it on the heptagon and every polygon opens up.

## Mistakes To Watch For With Heptagons
### Mistake 1: Dividing 360° instead of 900° for the interior angle
**Where it slips in:** finding a single interior angle of a regular heptagon. **Don't do this:** compute 360°÷7≈51.43° and call it the interior angle. **The correct way:** divide the interior _sum_ by 7: 900°÷7≈128.57°. Students first meeting polygon angles often reach for 360° because it is the "angles around a point" number, but that gives the exterior angle, not the interior one.
### Mistake 2: Using the regular-heptagon angle on an irregular heptagon
**Where it slips in:** angle problems where the heptagon is drawn lopsided. **Don't do this:** assume every angle is 128.57° when the sides are clearly unequal. **The correct way:** 128.57° applies _only_ to a regular heptagon. For an irregular one, the angles still sum to 900°, but individual angles vary, so you solve for the missing angle from that total.
### Mistake 3: Miscounting the number of triangles when deriving the angle sum
**Where it slips in:** re-deriving (n−2)×180° from scratch. **Don't do this:** cut a heptagon into 7 triangles (one per side) and get 7×180°=1260°. **The correct way:** cut from a _single vertex_, which gives n−2=5 triangles, not 7, so the sum is 900°. Getting the triangle count wrong here is the geometry version of an off-by-one error.

## Key Takeaways
- A **heptagon** is a **7-sided polygon** with 7 vertices and 7 interior angles.
- The interior angles always **sum to 900°**, from (n−2)×180° with n=7.
- A **regular heptagon** has each interior angle ≈128.57° and each exterior angle ≈51.43°.
- A heptagon has **14 diagonals** and, when regular, **7 lines of symmetry**.
- Regular heptagons **cannot tessellate** a flat plane.