Pentagon Calculator
Calculate area, perimeter, diagonal, apothem, and circumradius of a regular pentagon.
Pentagon details
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Measurement
Side length (s) ?
cm
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Anatomy of a regular pentagon
Hover any dimension to see the matching value. A regular pentagon is determined by a single length — every other quantity falls out of the same formulas.
Formula
A
=
5 s2
4
·
cot(π/5)
P
=
5s
a
=
s
2 tan(π/5)
R
=
s
2 sin(π/5)
d
=
s · φ
- A
- Area of the pentagon
- P
- Perimeter — the sum of all five sides
- s
- Side length — all five sides are equal
- a
- Apothem — centre to midpoint of a side (inscribed-circle radius)
- R
- Circumradius — centre to any vertex (circumscribed-circle radius)
- d
- Diagonal — joining any two non-adjacent vertices
- φ
- The golden ratio, (1 + √5) / 2 ≈ 1.61803
Worked example — your numbers
- s = —
- s² = —
- Area = (5 · s²/4) · cot(π/5) = —
- Perimeter = 5s = —
- Apothem = s / (2 tan(π/5)) = —
- Circumradius = s / (2 sin(π/5)) = —
- Diagonal = s · φ = —
The interior angle is always 108° — the exact value of (n−2) · 180° / n when n = 5. Scale the pentagon up or down and every angle stays the same; only the lengths grow. The diagonal-to-side ratio is exactly the golden ratio φ, which is why pentagons show up everywhere in classical geometry.