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Heptagon — Definition, Properties & Examples

Heptagon

A polygon with seven sides. Some authors also use the name septagon instead of heptagon.

 

An irregular heptagon (seven-sided polygon) with no labels, showing unequal sides and angles.

Heptagon

A regular heptagon (7-sided polygon) with equal sides and angles, no labels visible.

Regular Heptagon

Key Formula

S=(n−2)×180°S = (n - 2) \times 180°
Where:
  • SS = Sum of the interior angles of the polygon
  • nn = Number of sides (for a heptagon, n = 7)

Worked Example

Problem: Find the sum of the interior angles of a heptagon, and determine the measure of each interior angle if the heptagon is regular.
Step 1: Use the interior angle sum formula with n = 7.
S=(7−2)×180°S = (7 - 2) \times 180°
Step 2: Simplify inside the parentheses.
S=5×180°=900°S = 5 \times 180° = 900°
Step 3: For a regular heptagon, all seven angles are equal. Divide the total by 7.
Each angle=900°7≈128.57°\text{Each angle} = \frac{900°}{7} \approx 128.57°
Answer: The sum of the interior angles of any heptagon is 900°. In a regular heptagon, each interior angle measures approximately 128.57°.

Another Example

Problem: A regular heptagon has a side length of 10 cm. Find the measure of each exterior angle and the total number of diagonals.
Step 1: Each exterior angle of a regular polygon equals 360° divided by the number of sides.
Exterior angle=360°7≈51.43°\text{Exterior angle} = \frac{360°}{7} \approx 51.43°
Step 2: Use the diagonal formula for a polygon with n sides.
d=n(n−3)2=7(7−3)2=7×42=14d = \frac{n(n-3)}{2} = \frac{7(7-3)}{2} = \frac{7 \times 4}{2} = 14
Answer: Each exterior angle of a regular heptagon is approximately 51.43°, and a heptagon has 14 diagonals.

Frequently Asked Questions

How many sides does a heptagon have?
A heptagon has exactly 7 sides. The name comes from the Greek word 'hepta,' meaning seven. It also has 7 vertices and 7 interior angles.
What is the difference between a heptagon and a septagon?
They are the same shape — both refer to a 7-sided polygon. 'Heptagon' comes from Greek ('hepta' = seven), while 'septagon' comes from Latin ('septem' = seven). Heptagon is the more widely used term in mathematics.

Hexagon vs. Heptagon

A hexagon has 6 sides and interior angles summing to 720°, while a heptagon has 7 sides and interior angles summing to 900°. A regular hexagon has 60° exterior angles and tiles the plane perfectly; a regular heptagon has approximately 51.43° exterior angles and cannot tile the plane by itself.

Why It Matters

Heptagons appear in architecture, coin design (such as the UK 50-pence and 20-pence coins, which use curved heptagonal shapes), and tiling patterns. Understanding the heptagon also reinforces the general polygon angle formulas, since it is one of the simplest polygons whose regular form cannot be constructed exactly with a compass and straightedge alone.

Common Mistakes

Mistake: Confusing a heptagon (7 sides) with a hexagon (6 sides) or an octagon (8 sides).
Correction: Remember the Greek prefix: 'hepta' means seven. Hexagons have 6 sides and octagons have 8. The sequence is hex (6), hept (7), oct (8).
Mistake: Assuming each interior angle of a regular heptagon is a whole number of degrees.
Correction: Since 900° ÷ 7 is not a whole number, each interior angle is the irrational value 900°/7 ≈ 128.57°. Not every regular polygon has neat angle measures.

Related Terms

  • Polygon — General term for any closed multi-sided figure
  • Side of a Polygon — Each of the seven line segments of a heptagon
  • Regular Polygon — A heptagon with all equal sides and angles
  • Hexagon — A 6-sided polygon, one fewer side
  • Octagon — An 8-sided polygon, one more side
  • Interior Angle — Angle inside the polygon at each vertex
  • Diagonal — A heptagon has 14 diagonals