43.0119
sq units
25
units
3.441
units
8.0902
units
108
°
43.0119
sq units
25
units
3.441
units
8.0902
units
108
°
The regular pentagon—a five-sided polygon with equal sides and equal angles—holds a unique place in mathematics, art, and nature. Its geometry is intimately connected to the golden ratio φ = (1 + √5)/2 ≈ 1.6180, a constant that appears in the proportions of the pentagon's diagonals and sides. This deep relationship made the pentagon a symbol of the Pythagorean brotherhood and has fascinated mathematicians for over two millennia.
The Pentagon Calculator computes the essential geometric properties of a regular pentagon from a single measurement: the side length. It returns the area, perimeter, apothem (the distance from the center to the midpoint of a side), the diagonal length, and the interior angle. These calculations are valuable in architecture, design, tiling, and any discipline where pentagonal geometry appears.
The area of a regular pentagon is given by the formula A = (s²/4)√(25 + 10√5), which simplifies the general regular polygon area formula for n = 5. This expression emerges from trigonometric identities involving tan(36°) and can also be written as A = (5s²)/(4 tan(π/5)). The numerical coefficient √(25 + 10√5)/4 ≈ 1.72048 means a pentagon with side 1 has area approximately 1.72 square units.
The perimeter is simply five times the side length, and each interior angle of a regular pentagon measures exactly 108°. The exterior angle is 72°, and the sum of all interior angles is 540° = (5 − 2) × 180°. These angular properties mean that regular pentagons cannot tile the plane by themselves—unlike equilateral triangles, squares, and regular hexagons—because 108° does not evenly divide 360°.
The diagonal of a regular pentagon has length d = s · φ, where φ is the golden ratio. Moreover, the diagonals of a pentagon intersect to form a smaller regular pentagon inside, and the ratio of the diagonal to the side of this inner pentagon is again φ. This self-similar, recursive structure is the geometric basis for the Penrose tiling, a famous aperiodic tiling discovered by Roger Penrose in the 1970s that exhibits five-fold symmetry without repeating.
Pentagons appear throughout the natural world and human engineering. The cross-section of an okra pod, the arrangement of five petals in many flowers (roses, cherry blossoms), and the shape of starfish all reflect pentagonal symmetry. The Pentagon building in Arlington, Virginia—headquarters of the United States Department of Defense—is the world’s largest office building by floor area and is shaped as a regular pentagon with side length of approximately 281 meters.
In construction and design, pentagonal shapes appear in decorative tiles, window designs, and architectural motifs. The apothem is particularly useful because it represents the radius of the inscribed circle and determines how a pentagonal shape fits within rectangular or circular constraints. This calculator provides all these measurements instantly, making it an essential tool for students, engineers, designers, and mathematicians working with pentagonal geometry.
Given side length s of a regular pentagon, the calculator applies the following formulas:
$$A = \frac{s^2}{4}\sqrt{25 + 10\sqrt{5}}$$
$$P = 5s$$
$$a = \frac{s}{2\tan\left(\frac{\pi}{5}\right)}$$
$$d = s \cdot \varphi = s \cdot \frac{1+\sqrt{5}}{2}$$
$$\theta = 108°$$
The area formula is derived from decomposing the pentagon into five isosceles triangles meeting at the center. Each triangle has base s and height equal to the apothem. The golden ratio φ enters through the identity 2 cos(36°) = φ, linking the pentagon's trigonometry to this famous constant.
The area measures the total surface enclosed by the pentagon. The perimeter is the total edge length. The apothem is the shortest distance from the center to any edge and equals the inscribed circle’s radius. The diagonal connects two non-adjacent vertices and equals the side multiplied by the golden ratio. The interior angle of 108° is fixed for all regular pentagons regardless of size.
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A regular pentagon with side 10 encloses approximately 172.05 square units. The diagonal is about 16.18, which is 10φ.
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Scaling the side to 3.5 gives area ≈ 21.08 sq units, showing area scales as the square of the side length.
The golden ratio φ = (1+√5)/2 ≈ 1.618 is fundamental to pentagonal geometry. The diagonal-to-side ratio of a regular pentagon equals φ exactly. Additionally, the diagonals intersect each other in the golden ratio, and recursive subdivisions of the pentagon produce self-similar structures governed by φ.
For regular polygons to tile the plane, their interior angle must evenly divide 360°. The pentagon's interior angle of 108° does not divide 360° evenly (360/108 ≈ 3.33), so gaps or overlaps are unavoidable. Only equilateral triangles (60°), squares (90°), and regular hexagons (120°) can tile the plane among regular polygons.
Divide the regular pentagon into five congruent isosceles triangles from the center. Each has base s and height equal to the apothem a = s/(2 tan 36°). The total area is 5 × (1/2) × s × a = 5s²/(4 tan 36°). Using the identity tan 36° = √(5−2√5), this simplifies to s²√(25+10√5)/4.
The apothem is the radius of the inscribed circle—the largest circle that fits inside the pentagon touching all five sides. It is used to calculate area (A = ½Pa), to determine clearance for pentagonal shapes fitting inside circular constraints, and in engineering for bolt-circle and fitting calculations.
The circumradius (distance from center to vertex) is R = s/(2 sin(π/5)) = s/(2 sin 36°) ≈ 0.851s. This is the radius of the circumscribed circle that passes through all five vertices. You can also compute it as R = apothem / cos(π/5).
Five-fold symmetry appears in many flowers (roses, morning glories), starfish, sand dollars, and the cross-sections of some fruits (apples, okra). Viral capsids often display icosahedral symmetry with pentagonal faces. The prevalence of pentagonal symmetry in biology is related to optimal packing and growth patterns governed by the golden ratio.
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