Pentagon Calculator
Calculate the area, perimeter, apothem, diagonal, circumradius, and inradius of a regular pentagon. Instant results with step-by-step formulas.
Pentagon Calculator
Calculator
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Formula: A = (sqrt(5(5 + 2*sqrt(5))) / 4) * s^2
Worked example — Perimeter: 40 cm | Area: 110.11 cm^2 | Apothem: 5.5055 cm | Diagonal: 12.9443 cm
Formula
A = (sqrt(5(5 + 2*sqrt(5))) / 4) * s^2
Where A = Area of the regular pentagon, s = side length. The perimeter is P = 5s, the apothem is a = s / (2*tan(pi/5)), and the diagonal is d = s * (1+sqrt(5))/2 (the golden ratio times s).
Worked Examples
Example 1: Pentagon with Side Length 8 cm
Problem:Find the area, perimeter, apothem, and diagonal of a regular pentagon with a side length of 8 cm.
Solution:Perimeter = 5 * 8 = 40 cm Apothem = 8 / (2 * tan(pi/5)) = 8 / (2 * 0.7265) = 5.5055 cm Area = (1/2) * 40 * 5.5055 = 110.11 cm^2 Diagonal = 8 * (1 + sqrt(5)) / 2 = 8 * 1.6180 = 12.9443 cm
Result:Perimeter: 40 cm | Area: 110.11 cm^2 | Apothem: 5.5055 cm | Diagonal: 12.9443 cm
Example 2: Pentagon with Side Length 15 m
Problem:Calculate the circumradius, inradius, and area for a regular pentagon with side length 15 meters.
Solution:Circumradius = 15 / (2 * sin(pi/5)) = 15 / (2 * 0.58779) = 12.7598 m Inradius (apothem) = 15 / (2 * tan(pi/5)) = 15 / 1.4531 = 10.3228 m Area = (sqrt(5*(5+2*sqrt(5)))/4) * 15^2 = 1.72048 * 225 = 387.1080 m^2
Result:Circumradius: 12.7598 m | Inradius: 10.3228 m | Area: 387.11 m^2
Frequently Asked Questions
What is a regular pentagon and what are its key properties?
A regular pentagon is a five-sided polygon where all sides are equal in length and all interior angles are equal. Each interior angle of a regular pentagon measures exactly 108 degrees, and the sum of all interior angles totals 540 degrees. The pentagon has five lines of symmetry and rotational symmetry of order five. Regular pentagons appear frequently in nature and architecture, most famously in the shape of the United States Department of Defense headquarters building. The golden ratio (approximately 1.618) is intimately connected to the pentagon, as the ratio of a diagonal to a side equals the golden ratio.
How do you calculate the area of a regular pentagon?
The area of a regular pentagon with side length s is calculated using the formula A = (sqrt(5(5 + 2*sqrt(5))) / 4) * s^2, which simplifies to approximately A = 1.72048 * s^2. An alternative approach uses the apothem (the perpendicular distance from the center to any side): A = (1/2) * perimeter * apothem = (1/2) * 5s * a. The apothem itself is computed as a = s / (2 * tan(pi/5)). Both methods yield identical results. Understanding these formulas is essential for architects, engineers, and mathematicians who work with pentagonal shapes in design and construction projects.
What is the relationship between a pentagon and the golden ratio?
The golden ratio phi (approximately 1.6180339887) appears throughout the geometry of a regular pentagon. The diagonal of a regular pentagon divided by its side length equals the golden ratio exactly. Furthermore, the diagonals of a pentagon intersect each other in proportions governed by the golden ratio, creating a smaller regular pentagon inside. This self-similar property can repeat infinitely, producing a fractal-like pattern. The golden ratio connection makes pentagons significant in art, architecture, and design where aesthetically pleasing proportions are desired. The pentagram star formed by connecting all vertices also embodies the golden ratio in numerous ways.
What is the apothem and circumradius of a pentagon?
The apothem of a regular pentagon is the perpendicular distance from the center of the pentagon to the midpoint of any side. It is calculated as a = s / (2 * tan(pi/5)), where s is the side length. The circumradius is the distance from the center to any vertex, computed as R = s / (2 * sin(pi/5)). The apothem is also called the inradius because it equals the radius of the largest circle that fits inside the pentagon (inscribed circle). The circumradius is the radius of the smallest circle that completely contains the pentagon (circumscribed circle). These measurements are crucial for tiling, gear design, and structural engineering calculations.
How are pentagons used in real-world applications?
Pentagons have extensive real-world applications across many fields. In architecture, the Pentagon building in Washington D.C. is the most famous pentagonal structure in the world, and pentagonal floor tiles appear in various decorative patterns. In nature, many flowers have five petals arranged in pentagonal symmetry, including roses, buttercups, and morning glories. Starfish exhibit pentagonal radial symmetry. In sports, soccer balls use pentagons combined with hexagons to create a nearly spherical shape. In mathematics and crystallography, Penrose tilings use pentagons to create non-periodic tessellations. Engineers use pentagonal cross-sections in certain specialized structural components and fasteners.
Can a regular pentagon be constructed with compass and straightedge?
Yes, and it is one of the classical constructions in Book IV of Euclid Elements. It is possible because 5 is a Fermat prime, so the cosine of the central angle is expressible in square roots: cos 72 degrees equals (sqrt(5) - 1)/4, about 0.30902. Practically, the construction reduces to producing a segment in the golden ratio and stepping it around a circle. The related side length is s = 2R sin 36 degrees, where R is the circumradius, which is the numerical form Pentagon Calculator inverts when you supply a side.
Why can regular pentagons not tile a plane on their own?
Tiling without gaps requires the angles meeting at each vertex to total exactly 360 degrees. The interior angle of a regular pentagon is 108 degrees, and 360 divided by 108 is 3.33, not a whole number: three pentagons meeting at a point cover 324 degrees and leave a 36 degree wedge. Only the equilateral triangle, square, and regular hexagon divide 360 evenly, so only they tile alone. Certain irregular pentagons do tile the plane, and pentagonal symmetry also appears in non-periodic Penrose tilings, but no arrangement of regular pentagons alone will do it.
How do you find pentagon area from the circumradius or apothem instead of the side?
From the circumradius R, the pentagon splits into five isosceles triangles with apex angle 72 degrees, giving Area = 2.5 R squared times sin 72 degrees, which is 2.37764 R squared. From the apothem a, the area is 5 a squared times tan 36 degrees, or 3.63271 a squared. Both agree with the side formula: a pentagon of side 15 has circumradius 12.7598 and apothem 10.3228, and all three expressions return about 387.11 square metres. Which form you use depends on whether the drawing gives you a bolt circle, a wrench flat, or an edge.
How many diagonals does a regular pentagon have, and what do they form?
A pentagon has n(n - 3)/2 diagonals, which for n equal to 5 gives exactly five, and in a regular pentagon all five are the same length, phi times the side. Drawn together they trace a pentagram, and each pair of diagonals cuts the other in the golden ratio, dividing a diagonal into pieces in the proportion 1 to 1.618. The small pentagon enclosed at the centre of the pentagram has a side of s divided by phi squared, about 0.382 of the original, so the whole figure can nest inside itself indefinitely.
What changes if the pentagon is irregular rather than regular?
Every formula on this page assumes all five sides and all five angles are equal, and none of them apply to an irregular pentagon: there is no single apothem or circumradius, and the constant 1.72048 is meaningless. For an irregular pentagon, use the shoelace formula on the five vertex coordinates, or split the shape into three triangles from one vertex and add their areas. Only the interior angle sum of 540 degrees carries over, because that depends on the number of sides alone and not on the shape.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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