Pentagon Calculator - Area, Perimeter & Apothem
Calculate all properties of a regular pentagon from any single known value — enter the side length, apothem, area, or perimeter and get every other measurement instantly.
Select which property you already know, enter its value, and the calculator will compute the side length, perimeter, area, apothem, and diagonal.
Pentagon Calculator - Area, Perimeter & Apothem
Calculate all properties of a regular pentagon from any single known value — enter the side length, apothem, area, or perimeter and get every other measurement instantly.
About the Pentagon Calculator
A regular pentagon is a convex polygon with five equal sides and five equal interior angles. Each interior angle measures 108°, the sum of all interior angles is 540°, and the shape has five lines of reflective symmetry as well as rotational symmetry of order 5. Regular pentagons appear throughout nature — the cross-section of okra, the arrangement of petals in many flowers, and the symmetry of starfish arms all exhibit pentagonal geometry — and in art and architecture, from the Pentagon building in Washington, D.C., to decorative tiles and geodesic dome panels.
All properties of a regular pentagon derive from its side length s. The perimeter is simply P = 5s. The apothem a — the perpendicular distance from the center to the midpoint of any side — equals s / (2 × tan(π/5)), where tan(π/5) ≈ 0.7265. The area can be written in terms of the side length as A = (5s²) / (4 × tan(π/5)), or equivalently A = (5/2) × s × a using the apothem. The diagonal d (the distance between two non-adjacent vertices) is d = s × φ, where φ = (1 + √5) / 2 ≈ 1.6180 is the golden ratio — a beautiful connection that makes the regular pentagon intimately tied to the most famous irrational number in mathematics.
Because all four main properties (side, perimeter, area, apothem, and diagonal) are related through these formulas, knowing any one of them is enough to determine all the others. The inverse relationships are equally straightforward: given the area A, you can recover the side as s = √(4A × tan(π/5) / 5); given the apothem a, the side is s = 2a × tan(π/5); given the perimeter P, the side is s = P/5; given the diagonal d, the side is s = d/φ.
The golden ratio φ woven through pentagon geometry has fascinated mathematicians and artists for millennia. The ratio of the diagonal to the side of a regular pentagon is exactly φ, and the diagonals of a regular pentagon intersect each other in the golden ratio. Constructing a regular pentagon with compass and straightedge — a classical Euclidean construction — relies on constructing the golden ratio first. These properties made the pentagram (the five-pointed star formed by connecting every other vertex) a symbol of beauty and harmony in ancient Greek and Renaissance traditions.
Whether you are designing a pentagonal room, tiling a floor, calculating the floor area of a pentagonal building, cutting a five-sided piece of material, or exploring the golden ratio in mathematical artwork, this calculator lets you start from whichever measurement you have and instantly obtain every other property of the pentagon to the precision you need.
Pentagon Calculator Examples
Examples starting from each of the four common known values.
| Known value | Key results | Formula used |
|---|---|---|
| Side length s = 10 | Perimeter = 50, Area ≈ 172.05, Apothem ≈ 6.882, Diagonal ≈ 16.18 | A = 5s²/(4 tan(π/5)); diagonal = s × φ ≈ 1.618 × 10. |
| Apothem a = 6.882 | Side ≈ 10.00, Perimeter ≈ 50.00, Area ≈ 172.05, Diagonal ≈ 16.18 | s = 2a × tan(π/5) ≈ 2 × 6.882 × 0.7265. |
| Area = 172.05 | Side ≈ 10.00, Perimeter ≈ 50.00, Apothem ≈ 6.882, Diagonal ≈ 16.18 | s = √(4A × tan(π/5) / 5). |
| Perimeter = 50 | Side = 10, Area ≈ 172.05, Apothem ≈ 6.882, Diagonal ≈ 16.18 | s = P/5 = 10; all other properties follow. |
How to Use the Pentagon Calculator
- Select the property you already know from the Calculate From dropdown: Side Length, Apothem, Area, Perimeter, or Diagonal.
- Enter the known value in the Value field with the appropriate unit (the output will use the same unit).
- Click Calculate. The calculator applies the relevant pentagon formula and displays all five properties instantly.
- Review the side length, perimeter, area, apothem, and diagonal in the result panel.
- Click Reset to clear the fields and calculate for a different pentagon.
Pentagon Calculator FAQ
What is the formula for the area of a regular pentagon?
The area of a regular pentagon with side length s is A = (5s²) / (4 × tan(π/5)). Numerically, tan(π/5) ≈ 0.72654, so A ≈ 1.72048 × s². You can also write it as A = (5/2) × s × a, where a is the apothem — this is the standard polygon area formula: half the perimeter times the apothem.
What is the apothem of a pentagon?
The apothem is the perpendicular distance from the center of the pentagon to the midpoint of one of its sides. For a regular pentagon with side s, the apothem is a = s / (2 × tan(π/5)) ≈ 0.6882 × s. It is also the inradius — the radius of the largest circle that fits inside the pentagon touching all five sides.
Why does the diagonal equal side times the golden ratio?
In a regular pentagon, the diagonal d and side s satisfy d/s = φ = (1 + √5)/2 ≈ 1.618. This emerges from the geometry of the isosceles triangle formed by two sides and a diagonal: the ratio of its longer to shorter sides is exactly φ. It is one of the most elegant appearances of the golden ratio in elementary geometry.
What is the perimeter of a regular pentagon?
The perimeter of a regular pentagon is P = 5 × s, where s is the side length. Because all five sides are equal, you simply multiply the side length by five. Conversely, if you know the perimeter, divide by five to find the side length.
How many diagonals does a regular pentagon have?
A regular pentagon has exactly five diagonals — each vertex connects to the two non-adjacent vertices. The five diagonals form a regular pentagram (five-pointed star) inside the pentagon. Every diagonal has the same length d = s × φ, and the diagonals intersect each other in the golden ratio.
Can this calculator work with different units?
Yes. The calculator works with any consistent unit — centimetres, metres, inches, feet, or any other linear unit. Just enter the known value in your chosen unit, and all output measurements will be in the same unit (with area in the square of that unit, e.g. cm² if you entered cm).