What Is a Heptagon?
A heptagon (also called a septagon) is a polygon with seven sides and seven angles. A regular heptagon has all sides of equal length and all interior angles equal. Each interior angle of a regular heptagon measures approximately 128.571 degrees (900/7 degrees).
Unlike the triangle, square, or hexagon, a regular heptagon cannot be constructed with just a compass and straightedge. It is one of the simplest regular polygons that require more advanced construction methods.
Heptagon Area Formulas
From Side Length
The most common method using the side length of the regular heptagon.
From Apothem
Using the apothem (distance from center to the midpoint of a side).
From Circumradius
Using the circumradius (distance from center to a vertex).
Key Properties of a Regular Heptagon
- Number of sides: 7
- Sum of interior angles: 900 degrees
- Each interior angle: 128.571 degrees (900/7)
- Each exterior angle: 51.429 degrees (360/7)
- Number of diagonals: 14
- Symmetry: 7 lines of symmetry and rotational symmetry of order 7
Relationship Between Side, Apothem, and Circumradius
For a regular heptagon with side length s:
- Apothem: a = s / (2 x tan(pi/7)) ≈ 1.0383 x s
- Circumradius: R = s / (2 x sin(pi/7)) ≈ 1.1524 x s
- Perimeter: P = 7s
- Area: A = (1/2) x P x a = (7/2) x s x a
Real-World Applications
Heptagonal shapes appear in architecture, coin design (the British 20p and 50p coins are heptagonal), and various engineering applications. The UK's 50-pence and 20-pence coins are notable examples of curves of constant width based on the heptagon, making them usable in vending machines despite not being circular.
Step-by-Step Example
Find the area of a regular heptagon with side length 10:
- Write the formula: A = (7/4) x s2 x cot(pi/7)
- Compute cot(pi/7) = cos(pi/7) / sin(pi/7) ≈ 2.0765
- Calculate s2 = 102 = 100
- Multiply: A = (7/4) x 100 x 2.0765 = 1.75 x 100 x 2.0765
- Result: A ≈ 363.391 square units