Use this regular pentagon calculator to quickly find all key measurements from one known value.
What this pentagon calculator does
A pentagon has five sides, but not every pentagon behaves the same way mathematically. This calculator is built for a regular pentagon, where all sides are equal and all interior angles are equal. If you provide one known measurement, the tool computes the rest instantly: side length, perimeter, area, apothem, circumradius, and diagonal.
This is useful in geometry homework, architectural sketches, CNC design, laser cutting layouts, and any project where consistent five-sided geometry matters.
Regular pentagon formulas used
Core relationships
- Perimeter: P = 5s
- Apothem: a = s / (2tan(π/5))
- Circumradius: R = s / (2sin(π/5))
- Area: A = (P × a) / 2 = (5s²) / (4tan(π/5))
- Diagonal: d = φs, where φ = (1 + √5)/2
Inverse relationships (to recover side length)
- From perimeter: s = P/5
- From area: s = √((4A tan(π/5))/5)
- From apothem: s = 2a tan(π/5)
- From circumradius: s = 2R sin(π/5)
- From diagonal: s = d/φ
How to use the calculator
- Select which measurement you already know.
- Enter the value as a positive number.
- Optionally add a unit label (like cm or m).
- Choose your preferred decimal precision.
- Click Calculate to get all related dimensions.
Worked example
Suppose the side length is 10 cm. The calculator finds:
- Perimeter = 50 cm
- Apothem ≈ 6.8819 cm
- Circumradius ≈ 8.5065 cm
- Diagonal ≈ 16.1803 cm
- Area ≈ 172.0477 cm²
This gives you every major measurement from one input, with no manual trigonometry.
Regular vs. irregular pentagons
If your shape is irregular (different side lengths or angles), these formulas do not apply directly. In that case, you usually need to split the polygon into triangles and use coordinate geometry or triangulation methods. This calculator is intentionally focused on the regular case for speed and reliability.
Common mistakes to avoid
- Entering negative or zero values.
- Using this tool for an irregular pentagon.
- Mixing units (for example, entering mm but interpreting results as cm).
- Rounding too aggressively in intermediate manual calculations.
Final notes
A regular pentagon has fixed angle properties: each interior angle is 108°, and each central angle is 72°. This geometric consistency is exactly why a pentagon calculator is so efficient: one reliable measurement can unlock the rest.