Heptagon Calculator

Calculate area, perimeter, apothem, diagonals, and angles of a regular heptagon (7-sided polygon).

Enter Side Length

Result

Area
0
square units
Side Length (s)
Perimeter (P)
Area (A)
Apothem (a)
Circumradius (R)
Interior Angle
Exterior Angle
Number of Diagonals
Short Diagonal
Long Diagonal

Step-by-Step Solution

A = (7/4) * s^2 * cot(pi/7)

Understanding Regular Heptagons

A regular heptagon (also called a septagon) is a polygon with seven equal sides and seven equal angles. It is one of the more fascinating regular polygons because, unlike the triangle, square, and hexagon, a regular heptagon cannot be constructed with a compass and straightedge alone. Each interior angle of a regular heptagon measures approximately 128.571 degrees (exactly 900/7 degrees).

Heptagon Formulas

Area

The area in terms of side length s using the cotangent formula.

A = (7/4) * s2 * cot(π/7)

Perimeter

Simply seven times the side length.

P = 7s

Apothem

The distance from center to midpoint of a side.

a = s / (2 * tan(π/7))

Circumradius

The distance from center to a vertex.

R = s / (2 * sin(π/7))

Interior Angle

Each interior angle in a regular heptagon.

θ = (7-2)*180/7 = 128.571...°

Number of Diagonals

Total diagonals in any n-sided polygon.

d = n(n-3)/2 = 7(4)/2 = 14

Properties of a Regular Heptagon

  • Sides: 7 equal sides
  • Vertices: 7
  • Diagonals: 14 (two distinct lengths: short and long)
  • Interior angle sum: (7-2) × 180 = 900 degrees
  • Each interior angle: 900/7 ≈ 128.571 degrees
  • Each exterior angle: 360/7 ≈ 51.429 degrees
  • Symmetry: 7 lines of symmetry and rotational symmetry of order 7

Diagonal Lengths

A regular heptagon has 14 diagonals, which come in two distinct lengths. The shorter diagonal connects vertices separated by one vertex, while the longer diagonal connects vertices separated by two vertices. These diagonals have lengths related to the side length s by factors involving trigonometric functions of multiples of π/7.

Historical and Cultural Significance

  • Architecture: Heptagonal floor plans appear in some historical buildings and defensive fortifications.
  • Coins: Several countries use heptagonal coins (e.g., the UK 50p and 20p coins are equilateral-curve heptagons).
  • Mathematics: The impossibility of constructing a regular heptagon with compass and straightedge was proven using Galois theory.
  • Nature: Some flowers and seed patterns exhibit seven-fold symmetry.

Area Calculation Methods

The area of a regular heptagon can be calculated several ways: using the side length with A = (7/4)s2cot(π/7), using the apothem with A = (1/2) × perimeter × apothem, or using the circumradius with A = (7/2)R2sin(2π/7). All methods yield the same result for a given heptagon.