What Is A Heptagon?
A heptagon is a closed, flat shape (a polygon) 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 $180°$.
$$\text{Sum of interior angles} = (n - 2) \times 180°$$
For a heptagon, $n = 7$:
$$(7 - 2) \times 180° = 5 \times 180° = 900°$$
For a regular heptagon, share that total equally across the 7 angles:
$$\text{Each interior angle} = \frac{900°}{7} \approx 128.57°$$
The exterior angles of any polygon always sum to $360°$, so each exterior angle of a regular heptagon is
$$\frac{360°}{7} \approx 51.43°$$
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 | $\approx 128.57°$ |
Each exterior angle | $\approx 51.43°$ |
Number of diagonals | 14 |
Lines of symmetry | 7 |
The number of diagonals uses $\frac{n(n-3)}{2} = \frac{7 \times 4}{2} = 14$, and the full interior angles rule generalises 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 $n = 7$.
$$(n - 2) \times 180° = (7 - 2) \times 180°$$
$$= 5 \times 180° = 900°$$
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": $360° \div 7 \approx 51.43°$.
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:
$$\text{Each interior angle} = \frac{900°}{7} \approx 128.57°$$
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 $900°$.
$$130° + 140° + 120° + 150° + 100° + 110° = 750°$$
$$\text{Seventh angle} = 900° - 750° = 150°$$
Example 4
How many diagonals does a heptagon have?
Use the diagonal formula, where $n$ is the number of sides.
$$\frac{n(n - 3)}{2} = \frac{7(7 - 3)}{2} = \frac{7 \times 4}{2} = 14$$
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.
$$\text{Each exterior angle} = \frac{360°}{7} \approx 51.43°$$
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 $\approx 128.57°$. Try fitting them around one point:
$$2 \times 128.57° = 257.14° \quad (\text{leaves a gap})$$
$$3 \times 128.57° = 385.71° \quad (\text{overlaps, exceeds } 360°)$$
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 $(n-2) \times 180°$ rule you used here scales to every polygon - the octagon, the decagon, and beyond - and it is the same rule that fixes the angles of the quadrilaterals you already know. Master it on the heptagon and every polygon opens up.
Because the heptagon has a constant width when drawn as a Reuleaux-style curve, real coins like the 50-pence piece use a seven-sided shape so vending machines can measure them by width from any angle.
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 $360° \div 7 \approx 51.43°$ and call it the interior angle.
The correct way: divide the interior sum by 7: $900° \div 7 \approx 128.57°$. 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 $(n-2) \times 180°$ from scratch.
Don't do this: cut a heptagon into 7 triangles (one per side) and get $7 \times 180° = 1260°$.
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 - the same slip that caused the Hyatt Regency walkway collapse in 1981, where a single miscounted load path doubled the force on a connection and killed 114 people. Count the pieces carefully; the setup matters more than the arithmetic.
Key Takeaways
A heptagon is a 7-sided polygon with 7 vertices and 7 interior angles.
The interior angles always sum to 900°, from $(n - 2) \times 180°$ with $n = 7$.
A regular heptagon has each interior angle $\approx 128.57°$ and each exterior angle $\approx 51.43°$.
A heptagon has 14 diagonals and, when regular, 7 lines of symmetry.
Regular heptagons cannot tessellate a flat plane.
To take heptagons further with a teacher, explore Bhanzu's geometry tutor sessions, a middle school math tutor for polygons and angles, or general math classes online.
A Practical Next Step
Practice these problems to solidify your understanding. Work through them and check the answers below.
Find each interior angle of a regular heptagon (round to two decimal places).
An irregular heptagon has six angles: 120°, 135°, 140°, 100°, 155°, and 130°. Find the seventh.
How many diagonals does a heptagon have?
Answer to Question 1: $900° \div 7 \approx 128.57°$. Answer to Question 2: $900° - 780° = 120°$. Answer to Question 3: $\frac{7 \times 4}{2} = 14$ diagonals.
Want a live Bhanzu trainer to walk through more heptagon problems? Book a free demo class.
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