Pentagon Calculator
Free Pentagon Calculator for 2d geometry. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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 * 0.5878) = 12.7627 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: 12.7627 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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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