Decagon Calculator

Calculate the area, perimeter, apothem, diagonals, and angles of a regular decagon from the side length.

Enter Decagon Dimensions

Result

Area
--
square units
Perimeter --
Apothem --
Circumradius (R) --
Number of diagonals --
Interior angle --
Exterior angle --
Sum of interior angles --

Step-by-Step Solution

A = (5/2) x s^2 x sqrt(5 + 2sqrt(5))

What Is a Decagon?

A decagon is a polygon with exactly 10 sides and 10 angles. A regular decagon has all sides of equal length and all interior angles equal to 144 degrees. The decagon is one of the fundamental geometric shapes studied in mathematics and appears in various real-world applications, from architecture to coin design.

Properties of a Regular Decagon

A regular decagon has several distinctive properties that make it mathematically interesting:

  • All 10 sides are of equal length.
  • All 10 interior angles are equal, each measuring 144 degrees.
  • Each exterior angle measures 36 degrees.
  • The sum of all interior angles is 1,440 degrees.
  • It has 35 diagonals.
  • It has 10 lines of symmetry and rotational symmetry of order 10.

Decagon Formulas

Area

The area of a regular decagon from the side length s.

A = (5/2)s^2 sqrt(5+2sqrt(5))

Perimeter

The total length around the decagon.

P = 10s

Apothem

The distance from the center to the midpoint of any side.

a = s / (2 tan(pi/10))

Circumradius

The distance from the center to any vertex.

R = s / (2 sin(pi/10))

Interior Angle

Each interior angle of a regular decagon.

angle = (10-2) x 180 / 10 = 144 deg

Number of Diagonals

Total diagonals in a decagon.

d = 10(10-3)/2 = 35

The Golden Ratio Connection

The regular decagon has a fascinating connection to the golden ratio (phi = 1.618...). The ratio of the diagonal to the side length of a regular decagon equals the golden ratio. This relationship arises because the central angle of a regular decagon is 36 degrees, which is closely linked to the geometry of the regular pentagon and the golden ratio. The decagon can be constructed using a compass and straightedge, which is possible precisely because of this golden ratio connection.

Real-World Examples

  • Architecture: Decagonal floor plans and decorative elements in Islamic architecture and modern buildings.
  • Coins: Several countries use decagonal shapes for their coins, including the Australian 50-cent coin.
  • Engineering: Decagonal cross-sections in structural engineering and mechanical components.
  • Tiling: Decagonal symmetry appears in Penrose tilings and quasicrystals.
  • Military: Some historical fortifications used decagonal star-fort designs.

Constructing a Regular Decagon

A regular decagon can be constructed with a compass and straightedge. The construction relies on first creating a regular pentagon, since the central angle of a decagon (36 degrees) is exactly half that of a pentagon (72 degrees). By bisecting the arcs of an inscribed pentagon, you obtain the 10 equally-spaced vertices of a regular decagon.