Free Pentagon Calculator

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A regular pentagon has five equal sides and five equal interior angles (each 108°), with all its key measurements — area, apothem, and diagonal — following from the side length via closed-form formulas, several of which involve the golden ratio φ. Regular pentagons appear in architecture (the Pentagon building's footprint), in the classic five-pointed star (formed by drawing all the diagonals of a regular pentagon), and in the study of tiling and symmetry, since — unlike triangles, squares, and hexagons — a regular pentagon alone cannot tile a flat plane without gaps.

How it works

Enter the side length s. The area formula, Area=¼√(5(5+2√5))s², comes from splitting the pentagon into five congruent isosceles triangles from its center and summing their areas. The apothem — the distance from center to the midpoint of a side — is Apothem=s/(2tan(π/5)), derived from the right triangle formed by the apothem, half a side, and a line to a vertex. The diagonal (connecting two non-adjacent vertices) is Diagonal=s×φ, where φ=(1+√5)/2≈1.618 is the golden ratio.

Enter side length.
Click Calculate to see your results.

Examples

Side length 5

Area = ¼√(5(5+2√5))(25) ≈ 43.01. Apothem = 5/(2tan(36°)) ≈ 3.44. Diagonal = 5×1.618 ≈ 8.09.

Side length 10

Area = ¼√(5(5+2√5))(100) ≈ 172.05 (four times the area of the side-5 pentagon, since area scales with the square of the side length). Diagonal = 10×1.618 ≈ 16.18.

Common mistakes to avoid

Confusing the pentagon's apothem with its circumradius (center-to-vertex distance) — they are different lengths, with the circumradius always slightly longer.
Applying these formulas to an irregular pentagon (unequal sides or angles) — they are only valid for a fully regular pentagon.
Forgetting that area scales with the square of the side length, so doubling the side quadruples the area, not just doubles it.

Frequently asked questions

In a regular pentagon, the ratio of a diagonal to a side is always exactly φ ≈ 1.618 — one of the classic and most well-known appearances of the golden ratio in geometry, directly tied to the pentagon's 108° interior angles.

Drawing all five diagonals creates a five-pointed star (pentagram), with a smaller regular pentagon appearing in the middle — and that inner pentagon's diagonals would create an even smaller pentagram, in principle repeating forever.

No — a regular pentagon's interior angle (108°) doesn't divide evenly into 360°, so regular pentagons alone always leave gaps or overlaps when you try to tile a flat surface with them.

The general regular-polygon formula, Area = (1/2) × perimeter × apothem, applies here too: with perimeter = 5s and the apothem formula above, the two combine into the single-variable formula that only needs the side length.

540°, following the general polygon formula (n-2)×180° with n=5; dividing evenly among the five equal angles of a regular pentagon gives 108° per angle.

The hexagon has a larger area, since increasing the number of sides for a fixed side length brings a regular polygon closer to enclosing a circle, and each additional side adds proportionally more enclosed area.