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Regular Pentagon Calculator

Compute the area, perimeter, apothem, circumradius, and diagonal of any regular pentagon by entering a single side length.

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Regular Pentagon Calculator: Formulas and Methodology

A regular pentagon is a five-sided polygon in which all five sides share an identical length and all five interior angles each measure exactly 108°. The sum of the interior angles totals 540°. Because of its perfect five-fold rotational symmetry, the regular pentagon appears throughout mathematics, nature, and engineering — from the cross-section of starfruit to the famous five-sided footprint of the U.S. Department of Defense headquarters. This pentagon calculator applies five standard Euclidean formulas to a single input — side length s — to produce every key geometric measurement.

The Five Core Formulas

Every geometric property of a regular pentagon follows from a single measurement: the side length s. The formulas below, documented by Wolfram MathWorld's Regular Pentagon and Wikipedia's Pentagon article, cover all fundamental dimensions:

  • Area: A = (1/4) × √(5(5 + 2√5)) × s² ≈ 1.72048 × s²
  • Perimeter: P = 5s
  • Apothem (inradius): a = s / (2 × tan(π/5)) ≈ 0.68819 × s
  • Circumradius: R = s / (2 × sin(π/5)) ≈ 0.85065 × s
  • Diagonal: d = φ × s ≈ 1.61803 × s

The symbol φ (phi) denotes the golden ratio = (1 + √5)/2 ≈ 1.61803339, one of mathematics' most celebrated constants.

Variable Definitions

Side Length (s)

The side length is the straight-line distance between two adjacent vertices. Since all five sides of a regular pentagon are equal, s fully determines the shape's scale and every derived measurement.

Area (A)

The area is the total surface enclosed within the five sides. The irrational constant √(5(5 + 2√5)) / 4 evaluates numerically to approximately 1.72048. A regular pentagon with a side length of 10 cm encloses an area of approximately 172.05 cm².

Perimeter (P)

The perimeter equals the total boundary length, calculated as five times the side length. A pentagon with side 7 m has a perimeter of 35 m.

Apothem (a)

The apothem is the perpendicular distance from the geometric center to the midpoint of any side. It equals the radius of the incircle — the largest circle that fits inside the pentagon. For s = 10, a ≈ 6.882.

Circumradius (R)

The circumradius is the distance from the center to any vertex. It equals the radius of the circumcircle — the smallest circle that passes through all five vertices. For s = 10, R ≈ 8.507. The circumradius always exceeds the apothem because vertices project beyond the midpoints of the sides.

Diagonal (d)

The diagonal connects two non-adjacent vertices. A regular pentagon has exactly five diagonals of identical length. Their length equals the side length multiplied by the golden ratio: d = φs. For s = 10, d ≈ 16.180.

Deriving the Area Formula

Partitioning the pentagon into five congruent isosceles triangles, each with its apex at the center and its base along one side, gives a single-triangle area of (1/2) × s × a. Summing all five triangles yields A = (5/2) × s × a. Substituting a = s / (2 tan(π/5)) produces A = 5s² / (4 tan(π/5)). Applying the identity tan(36°) = √(5 − 2√5) and simplifying algebraically delivers the closed form A = (s²/4) √(5(5 + 2√5)).

The Golden Ratio and the Pentagon

The ratio of a regular pentagon's diagonal to its side length equals exactly φ = (1 + √5)/2 ≈ 1.61803, as explored at length in Wikipedia's Golden Ratio article. Every pair of intersecting diagonals divides each other in this same ratio, generating self-similar nested pentagons and pentagrams without end. This property makes the regular pentagon foundational in the study of quasicrystals, phyllotaxis, and classical architectural proportion.

Worked Example: Pentagon Ceramic Tile

A designer specifies regular pentagon tiles with a side length of 12 cm. Applying each formula:

  • Area: A ≈ 1.72048 × 144 ≈ 247.75 cm²
  • Perimeter: P = 5 × 12 = 60 cm
  • Apothem: a ≈ 0.68819 × 12 ≈ 8.26 cm
  • Circumradius: R ≈ 0.85065 × 12 ≈ 10.21 cm
  • Diagonal: d ≈ 1.61803 × 12 ≈ 19.42 cm

The apothem (8.26 cm) reveals the maximum circular inset that fits inside each tile. The circumradius (10.21 cm) determines the minimum circular blank required to cut the tile. The diagonal (19.42 cm) governs the spacing of star-pattern overlays. All five values follow from a single measurement with no iterative calculation required.

Interior Angles and Symmetry

Each interior angle of a regular pentagon measures (5 − 2) × 180° / 5 = 108°. Each exterior angle measures 72°. The pentagon belongs to the dihedral symmetry group D5, possessing five axes of reflective symmetry and five-fold rotational symmetry at multiples of 72°. These symmetry properties make the regular pentagon a fundamental object in tiling theory, molecular chemistry, and five-fold biological patterns.

Reference

Frequently asked questions

What is a regular pentagon and how does it differ from an irregular pentagon?
A regular pentagon has five equal side lengths and five equal interior angles, each measuring exactly 108°, with a total interior angle sum of 540°. An irregular pentagon also has five sides, but its sides and angles are not all equal. Only the regular pentagon exhibits perfect five-fold rotational symmetry, which allows the closed-form area, apothem, circumradius, and diagonal formulas used in this calculator to apply directly and without approximation.
How do you calculate the area of a regular pentagon from the side length?
Multiply the square of the side length by the constant 1.72048, which represents (1/4) × √(5(5 + 2√5)). For example, a regular pentagon with a side length of 5 cm has an area of 1.72048 × 25 ≈ 43.01 cm². The formula derives from dividing the pentagon into five congruent isosceles triangles from the center and summing their areas, with the apothem serving as each triangle's height.
What is the golden ratio and why does it appear in pentagon diagonal calculations?
The golden ratio φ = (1 + √5)/2 ≈ 1.61803 is an irrational constant that emerges directly from the geometry of the regular pentagon. The diagonal of any regular pentagon equals exactly φ times the side length. Furthermore, when two diagonals intersect inside the pentagon, they divide each other in the golden ratio. This intrinsic relationship makes the pentagon foundational to the mathematical study of φ in art, architecture, and nature.
What is the apothem of a regular pentagon and how does it differ from the circumradius?
The apothem is the perpendicular distance from the center to the midpoint of any side, equal to s / (2 × tan(36°)) ≈ 0.6882 × s. The circumradius is the distance from the center to any vertex, equal to s / (2 × sin(36°)) ≈ 0.8507 × s. The circumradius always exceeds the apothem because vertices project beyond side midpoints. For s = 10, the apothem is approximately 6.88 and the circumradius approximately 8.51, a ratio of about 1.236.
How many degrees is each interior angle of a regular pentagon?
Each interior angle of a regular pentagon measures exactly 108°. This follows from the general polygon interior angle formula: (n − 2) × 180° / n, where n = 5, giving (5 − 2) × 180° / 5 = 540° / 5 = 108°. The exterior angle at each vertex is 72°. The sum of all five interior angles equals 540°, a property that holds for every pentagon regardless of whether it is regular or irregular.
What are the most common real-world applications of regular pentagon geometry?
Regular pentagon geometry applies across architecture (five-sided building footprints and decorative tiling patterns), mechanical engineering (five-lobe bolt and knob designs), biology (five-petaled flowers, starfish, and sea urchin radial symmetry), materials science (quasicrystalline atomic lattices), and graphic design (five-pointed star construction). The pentagon calculator helps engineers verify circumscribed circle clearance for rotating components and helps designers determine inset circle diameters and diagonal spacing for decorative panels.