Calculate all properties of a regular pentagon from side length, area, perimeter, apothem, or circumradius. Includes area, perimeter, apothem, diagonals, circumradius, and golden ratio connections.
The regular pentagon — a five-sided polygon with all sides and angles equal — holds a special place in geometry thanks to its deep connection to the golden ratio φ = (1+√5)/2 ≈ 1.618. Every diagonal of a regular pentagon is exactly φ times the side length, and when those diagonals intersect, they divide each other in the golden ratio. This relationship has fascinated mathematicians since ancient Greece.
A regular pentagon has interior angles of 108° each (the supplement of 72°, the central angle). Its area formula, A = (√(25+10√5)/4)s² ≈ 1.72s², is more complex than simpler polygons but follows from dividing the pentagon into 5 isosceles triangles sharing a vertex at the center. Each triangle has a base equal to the side length and a height equal to the apothem.
The most famous pentagon in the world is the U.S. Department of Defense headquarters in Arlington, Virginia — each outer wall is 281 meters (921 feet) long. Soccer balls feature 12 regular pentagonal panels (black) alongside 20 hexagonal panels (white) in the classic truncated icosahedron design. The pentagram (five-pointed star) is formed by drawing all five diagonals of a regular pentagon.
This calculator works in reverse too: enter the area, perimeter, apothem, or circumradius, and it derives the side length and all other properties. The golden ratio section explains why φ appears everywhere in pentagon geometry.
Regular Pentagon — Area, Perimeter, Diagonals & Golden Ratio problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter solve from, unit, and it returns side length, area, perimeter, apothem (inradius) in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Area: A = (√(25+10√5)/4)s² ≈ 1.72048s² Perimeter: P = 5s Apothem (inradius): a = s/(2 tan(π/5)) Circumradius: R = s/(2 sin(π/5)) Diagonal: d = φs = ((1+√5)/2)s Interior angle: 108° Central angle: 72° Number of diagonals: 5
Result: Area ≈ 172.05 cm², Perimeter = 50 cm, Diagonal ≈ 16.18 cm
For a regular pentagon with side 10 cm: Area = (√(25+10√5)/4) × 100 ≈ 172.05 cm². Perimeter = 5 × 10 = 50 cm. Apothem ≈ 6.88 cm. Circumradius ≈ 8.51 cm. Diagonal = φ × 10 ≈ 16.18 cm (the golden ratio in action).
This regular pentagon — area, perimeter, diagonals & golden ratio tool links the entered values (solve from, unit) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Regular Pentagon — Area, Perimeter, Diagonals & Golden Ratio shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (side length, area, perimeter, apothem (inradius)) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
A = (√(25+10√5)/4)s², where s is the side length. This simplifies to approximately 1.72 × s². For side 10: A ≈ 172.05 square units.
The diagonal of a regular pentagon equals φ × side, where φ = (1+√5)/2 ≈ 1.618. When diagonals intersect, they divide each other in the ratio φ:1. The pentagram formed by the diagonals contains a smaller pentagon scaled by 1/φ².
108°. This comes from the formula (n−2)×180°/n = (5−2)×180°/5 = 540°/5 = 108°. The sum of all interior angles is 540°.
5 diagonals, from the formula n(n−3)/2 = 5(5−3)/2 = 5. All five diagonals are equal in a regular pentagon.
Regular pentagons cannot tile a plane by themselves (their 108° angle doesn't divide evenly into 360°). However, 15 types of irregular convex pentagons have been proven to tessellate, and many non-convex pentagons can too.
The apothem is the perpendicular distance from the center to the midpoint of a side. For a regular pentagon, it equals s/(2 tan(36°)) ≈ 0.688s. It's also the inradius (radius of the inscribed circle).