Pentagon Calculator
Calculate area, perimeter, side length, and apothem for a regular pentagon.
Calculation Results
Enter a value to compute all properties of a regular pentagon.
Side (a): units
Perimeter (P): units
Apothem (ap): units
Area (A): square units
Diagonal (d): units
Height (h): units
Circumcircle radius (R): units
Regular Pentagon Diagram
The Regular Pentagon
A regular pentagon is a 5-sided polygon with all sides and interior angles equal. Key relations:
- Perimeter: P = 5a
- Apothem: ap = a / (2·tan(π/5))
- Area: A = (5/2)·a·ap = (5a²)/(4·tan(π/5))
- Interior angle: 108° (each)
Side Length (s)
Apothem (a)
Perimeter (5s)
Area (A = ½ × P × a)
Pentagon Formulas
- Perimeter: P = 5a
- Apothem: ap = a / (2·tan(π/5))
- Area: A = (5/2)·a·ap = (5a²)/(4·tan(π/5))
Use radians for π/5; numerically, tan(π/5) ≈ 0.726542528.
How to Use the Pentagon Calculator
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Select What You Know
Choose one of: Side, Perimeter, or Area.
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Enter the Value
Provide a positive number and click Calculate.
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Read Results
The tool returns a, P, ap, and A. Use Reset to clear.
Example Calculation
- Given: a = 6 cm
- Perimeter: P = 5a = 30 cm
- Apothem: ap = 6 / (2·tan(π/5)) ≈ 4.132 cm
- Area: A = (5/2)·a·ap ≈ (2.5)·6·4.132 ≈ 61.98 cm²
Special Cases
- Regular Pentagon
P = 5a, ap = a/(2·tan(π/5)), A = (5a²)/(4·tan(π/5)).
- Irregular Pentagon
Perimeter = sum of sides. Area generally requires triangle decomposition or the coordinate shoelace formula; there isn't a single closed form in terms of one side.
This calculator assumes a regular pentagon.
Pentagon Formulas (Comparison)
| Type | Perimeter (P) | Apothem (ap) | Area (A) |
|---|---|---|---|
| Regular Pentagon | 5a | a / (2·tan(π/5)) | (5/2)·a·ap = (5a²)/(4·tan(π/5)) |
| Irregular Pentagon | a₁ + a₂ + a₃ + a₄ + a₅ | — | Decompose into triangles / shoelace |
Frequently Asked Questions
A regular pentagon is a five-sided polygon with all sides and interior angles equal.
For a regular pentagon, multiply the side length (s) by 5: P = 5s.
The apothem is the distance from the center of the pentagon to the midpoint of a side, perpendicular to the side.
Use the formula A = ½ × Perimeter × Apothem or A = (5/4) × s² × cot(π/5) for a regular pentagon.
Yes. For a regular pentagon: a = s / (2 × tan(π / 5)).
The calculator uses the same units as the input. If you enter side length in cm, the output will also be in cm² for area, etc.