Calculate area, perimeter, diagonals, and other properties of a regular pentagon.
Last updated: April 2026 | By Patchworkr Team
A regular pentagon is a five-sided polygon with all sides equal and all angles equal, each measuring 108°. The name comes from “penta” (five) and “gonia” (angle). The pentagon appears frequently in nature (starfish, flowers), art, and architecture. It holds special significance in mathematics because it relates to the golden ratio φ = (1 + √5)/2. Regular pentagons have five-fold rotational symmetry and five lines of reflectional symmetry, making them aesthetically pleasing and functionally important in design. The pentagon is the basis for creating pentagonal tilings and appears in molecular chemistry.
The mathematical properties of regular pentagons are fascinating and rich. The ratio of a pentagon’s diagonal to its side length is exactly the golden ratio, a relationship discovered by ancient mathematicians and revered ever since. This connection to the golden ratio makes pentagons central to understanding symmetry, proportion, and beauty in mathematics and nature. Understanding pentagons provides insight into regular polygons generally and introduces the crucial concept of the golden ratio, one of mathematics’ most important and ubiquitous constants.
All sides are equal in a regular pentagon
Why: Regularity means all sides are identical, so measuring just one side suffices and avoids averaging errors.
A = (s² √(25 + 10√5)) / 4
Why: This formula encodes the pentagon's unique geometry involving the golden ratio. It cannot be simplified further without losing precision.
P = 5s (multiply side by 5)
Why: Five equal sides means simply multiplying one side by 5 gives the total boundary length instantly.
Always 108° for regular pentagons
Why: Derived from (n-2)×180°/n formula where n=5. This constant angle makes pentagons predictable and useful for design.
d = s × (1 + √5) / 2 = s × φ
Why: This golden ratio relationship is unique to pentagons, making them mathematically special and appearing throughout nature and art.
Pentagon Architectural Design
The golden ratio φ ≈ 1.618 is the ratio of pentagon diagonal to side. It appears throughout nature and art.
Pentagons uniquely relate to the golden ratio, giving them special mathematical and aesthetic properties.
Each interior angle of a regular pentagon is 108°. All five sum to 540°.
No, regular pentagons cannot tile the plane by themselves, but they appear in semi-regular tilings.
In architecture, nature (flowers, starfish), design, and the 12-pentagon soccer ball pattern.
A five-pointed star is created by extending the sides of a pentagon.
Yes, the diagonal is longer by the golden ratio factor: d = φ × s.
The apothem is the distance from the center to the midpoint of a side (perpendicular distance).
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