Pentagon Calculator

Regular pentagon calculator

Tell the tool what you know — the side length, a diagonal, the circumradius, the inradius (apothem) or the area — and it derives every other property of a regular pentagon: area, perimeter, diagonal, circumradius and inradius. It is useful for geometry problems, tiling, drafting and design work involving five-sided shapes.

How it works

The calculator first converts your input into the side length, then applies the fixed ratios of a regular pentagon (φ is the golden ratio, (1+√5)/2 ≈ 1.618):

diagonal      = side × φ
circumradius  = side ÷ (2·sin 36°)   ≈ 0.851 × side
inradius      = side ÷ (2·tan 36°)   ≈ 0.688 × side
area          = (1/4)·√(5(5+2√5))·side²  ≈ 1.7205 × side²
perimeter     = 5 × side

If you enter the area instead of the side, it inverts the area formula (side = √(area ÷ 1.7205)) and the diagonal, radii and perimeter follow.

Example

A regular pentagon with side = 10:

• Perimeter: 5 × 10 = 50
• Diagonal: 10 × 1.618 = 16.18
• Circumradius: 10 × 0.851 = 8.51
• Inradius (apothem): 10 × 0.688 = 6.88
• Area: 1.7205 × 100 = 172.05

All calculations stay in your browser.

The golden ratio connection

One of the most striking properties of a regular pentagon is that its diagonal is exactly φ times the side length, where φ (phi) is the golden ratio (1 + √5) / 2 ≈ 1.6180. This is not a coincidence or approximation — it is an exact algebraic relationship that falls out of the pentagon’s 108° interior angles and the geometry of 72° isosceles triangles.

This means the diagonal-to-side ratio of a pentagon is the same proportion that appears in the Fibonacci sequence, in classical art and architecture, and in many natural growth patterns. A regular pentagon with a full set of diagonals drawn in creates a smaller regular pentagon at the center — and the ratio of the outer pentagon’s diagonal to the inner pentagon’s side is φ². This self-similar nesting continues indefinitely, which is why the pentagon is central to the construction of the regular pentagram.

Interior and central angles

Every regular polygon has two characteristic angles:

• Interior angle: the angle at each vertex, inside the shape. For a pentagon: (5 − 2) × 180° ÷ 5 = 108°. All five are equal.
• Central angle: the angle at the center subtended by each side, equal to 360° ÷ 5 = 72°.

The interior and central angles together always sum to 180°: 108 + 72 = 180. This is true for any regular polygon and reflects the relationship between the inscribed and circumscribed circles.

The 108° interior angle is why pentagons do not tile a flat plane by themselves — three pentagons meeting at a vertex would give 3 × 108° = 324°, leaving a 36° gap; four would give 432°, which exceeds 360°. This is in contrast to equilateral triangles, squares, and regular hexagons, which do tile perfectly.

Practical uses for pentagon calculations

Architecture and design. The Pentagon building in Washington is a real-world example: it has five sides of equal length with five internal courtyards arranged in a pentagonal pattern. Calculating the floor area, wall lengths, or courtyard dimensions of a pentagonal building requires exactly these formulas.

Tiling and craft. While pentagons alone do not tile the plane, they combine with other shapes in several tiling patterns. Knowing the inradius (apothem) is essential for calculating how pieces fit together in a composite tiling.

Art and sacred geometry. The pentagram — five diagonals of a regular pentagon — appears in classical and Renaissance design. Constructing one accurately requires knowing the exact diagonal length and the angles, which this calculator provides instantly for any size.

Mechanical design. Bolts, nuts, and fasteners occasionally appear in pentagonal profiles. Knowing the circumradius (the radius of the circle that encloses all five vertices) gives the minimum clearance needed for a pentagonal component.