What is the area formula for a regular pentagon?

The area of a regular pentagon with side length s is (s squared times the square root of 5 times (5 + 2 times the square root of 5)) divided by 4, which simplifies approximately to 1.7205 times s squared. Alternatively you can use area = 5 times s times apothem divided by 2, where the apothem is the perpendicular distance from the centre to the midpoint of a side.

What is the apothem of a pentagon?

The apothem of a regular polygon is the perpendicular distance from its centre to the midpoint of any side. For a regular pentagon with side s, the apothem equals s divided by (2 times tan(36 degrees)), which is approximately 0.6882 times s. The apothem is used in the standard area formula: area equals half the perimeter times the apothem.

What is the diagonal of a regular pentagon?

A regular pentagon has five diagonals, all of equal length. The diagonal length is the golden ratio phi (approximately 1.6180) times the side length s, so diagonal = s times (1 + sqrt(5)) / 2. The golden ratio appears naturally in regular pentagons, making them the basis for the construction of the icosahedron and dodecahedron in 3D geometry.

Pentagon Calculator

A regular pentagon is a five-sided polygon where all five sides are equal in length and all five interior angles are equal at 108 degrees each. It is one of the most geometrically rich simple shapes, connected intimately to the golden ratio: the diagonal of a regular pentagon divided by its side length is exactly phi = (1 + sqrt(5)) / 2, approximately 1.618. This relationship links the pentagon to Fibonacci numbers, icosahedra, and the way plants arrange leaves to maximise light capture. Regular pentagons also tile the faces of the dodecahedron, one of the five Platonic solids. This calculator computes every key measurement of a regular pentagon from just the side length: the area using the standard formula involving the square root of 5, the perimeter, the diagonal, the apothem (the perpendicular distance from the centre to a side, which is the inradius), the circumradius (the distance from the centre to a corner), and the interior angle. The default uses side = 4, giving area = 27.53, perimeter = 20, diagonal = 6.47, and apothem = 2.75.

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Side length (s)

units

27.53

area (square units)

Perimeter20.00

Diagonal6.47

Apothem2.75

Circumradius3.40

Interior angle of a regular pentagon is always 108 degrees. Apothem = inradius (centre to side midpoint).

How it works

For a regular pentagon with side length s: area = (s^2 / 4) * sqrt(5 * (5 + 2 * sqrt(5))), approximately 1.7205 * s^2. Perimeter = 5s. Diagonal = s * (1 + sqrt(5)) / 2 (the golden ratio times s). Apothem = s / (2 * tan(36 degrees)) = s * sqrt(5 + 2*sqrt(5)) / sqrt(20), approximately 0.6882 * s. Circumradius = s / (2 * sin(36 degrees)), approximately 0.8507 * s. Interior angle = (5 - 2) * 180 / 5 = 108 degrees.

Worked example

For side = 4: area = (16 / 4) * sqrt(5 * (5 + 2*sqrt(5))) = 4 * sqrt(25 + 10*sqrt(5)) = approximately 27.53. Perimeter = 5 * 4 = 20. Diagonal = 4 * 1.6180 = 6.47. Apothem = 4 / (2 * tan(36)) = 4 / (2 * 0.7265) = 2.75. Circumradius = 4 / (2 * sin(36)) = 4 / 1.1756 = 3.40. These match the defaults pre-filled above.

Related calculators

Rhombus Calculator: area, diagonals, and height of a rhombus.
Parallelogram Calculator: area and diagonals of a parallelogram.
Circle Segment Calculator: area of a circular segment.
Annulus Calculator: area between two concentric circles.

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