Decagon Area Calculator

Enter the side length of a regular decagon (10 equal sides) to get its area (A = (5/2)s²√(5+2√5)), perimeter, apothem, and circumradius.

Quick Facts

Area formula
A = (5/2)s²√(5+2√5) ≈ 7.6942s²
Sum of 10 congruent isosceles triangles meeting at the center.
Circumradius
R = s × φ ≈ 1.618034s
The center-to-vertex distance is always the side length times the golden ratio.
Interior angle
144° per vertex
All 10 interior angles of a regular decagon sum to 1440°.

Your Results

Calculated
Area
-
A = (5/2)s²√(5+2√5)
Perimeter
-
P = 10 × side
Apothem
-
a = s / (2 tan 18°)
Circumradius
-
R = s / (2 sin 18°) = s × φ

Ready

Enter a side length and unit, then press Calculate.

Formula and Method for the Area of a Regular Decagon

A regular decagon is a 10-sided polygon with all sides the same length and all 10 interior angles equal to 144°. Splitting it into 10 congruent isosceles triangles that meet at the center gives the exact area formula: A = (5/2)s²√(5+2√5), where s is the side length. This calculator also derives the perimeter, apothem, and circumradius from that same side length.

How the calculation works

Enter the side length and choose its unit. Internally the calculator first finds the apothem (the distance from the center to the midpoint of a side) using a = s / (2 tan 18°), and the circumradius (the distance from the center to a vertex) using R = s / (2 sin 18°). The area then follows from the general regular-polygon identity A = (1/2) × perimeter × apothem = 5 × s × a, which is algebraically identical to A = (5/2)s²√(5+2√5). Interestingly, the circumradius always works out to R = s × φ, where φ ≈ 1.618034 is the golden ratio — a direct consequence of the decagon's interior 36°/72° triangle geometry.

Common mistakes

  • Confusing side length with apothem or circumradius: the apothem (≈1.5388s) and circumradius (≈1.6180s) are both larger than the side length s — do not substitute one for another in the area formula.
  • Mixing units: keep the side length in one consistent unit throughout; convert inches to feet or centimeters to meters before entering the value.
  • Assuming regularity: these formulas only hold for a regular decagon (10 equal sides and equal angles). An irregular decagon needs coordinate-based methods, such as the shoelace formula.

Real-world applications

  • Decagonal tabletops, tiles, paving stones, and decorative panels use the area formula to estimate material coverage.
  • Architectural floor plans and gazebo or pavilion layouts use the perimeter and apothem to size decagonal footprints and roof framing.
  • Coin, medallion, and gasket design (some coins and mechanical parts use decagonal or near-decagonal outlines) relies on the circumradius to fit a shape within a given diameter.
  • Engineering and CAD work uses the apothem to check that a decagonal part clears or fits inside a circular bore or housing.

Frequently Asked Questions

What is the formula for the area of a regular decagon?
The area of a regular decagon (10 equal sides, each of length s) is A = (5/2)s²√(5+2√5) ≈ 7.6942 × s². This comes from splitting the decagon into 10 congruent isosceles triangles meeting at the center and summing their areas. A decagon with 6-inch sides has an area of about 277.0 in².
How do you find the apothem of a regular decagon?
The apothem (the distance from the center to the midpoint of a side) is a = s / (2 tan 18°) ≈ 1.53884 × s. The area can also be computed from the apothem using A = (1/2) × perimeter × apothem = 5 × s × a, which gives the same result as the direct area formula.
What is the circumradius of a regular decagon?
The circumradius (distance from the center to each vertex) is R = s / (2 sin 18°), which simplifies to R = s × φ, where φ is the golden ratio (≈1.618034). In other words, the distance from the center to a corner of a regular decagon is always the side length times the golden ratio.
Does this formula work for irregular decagons?
No. These formulas assume a regular decagon, where all 10 sides are equal and all 10 interior angles equal 144°. An irregular decagon has no single area formula — you would need to split it into triangles using its vertex coordinates and sum their areas (for example with the shoelace formula).

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