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Regular Pentagon Calculator: Area, Side & Diagonal
Enter one known measurement to calculate every dimension of a regular pentagon. Results and worked steps update as you type.
Enter one known measurement to calculate every dimension of a regular pentagon. Results and worked steps update as you type.
Known measurement
Live geometry diagram
Five isosceles center triangles. Schematic diagram; not to scale. Choose a result or a line in the diagram to use it as the known measurement.
a · Side lengthd · DiagonalR · Circumradiusr · Apothem
Results
How it was calculated
This calculator assumes five equal sides and five interior angles of 108°.
Golden ratio
d / a = φ = (1 + √5) / 2 ≈ 1.61803398875
The diagonal-to-side ratio is the golden ratio: d/a = φ ≈ 1.618.
A regular pentagon has only one diagonal length. The value a/φ ≈ 0.618a is not another diagonal.
Geometry formulas
P = 5a
A = 5ar / 2 = √[5(5 + 2√5)]a² / 4
d = φa
R = a / (2 sin 36°)
r = a / (2 tan 36°)
Regular pentagons cannot tile a flat plane without gaps or overlaps, so a hexagonal-tile estimate would be misleading.
Regular pentagon formulas
A regular pentagon has five equal sides and five equal angles. The calculator connects side length, perimeter, apothem, diagonal, and area.
- Perimeter = 5 x side length.
- Area = perimeter x apothem / 2.
- Diagonal = side x 1.61803398875.
Pentagon area calculator
Pentagon area can be calculated from side length, apothem, or perimeter depending on which measurement you know.
- Use apothem when the center-to-side distance is known.
- Use side length for most regular pentagon homework problems.
- Keep units consistent so area is in square units.
When to use it
Use this tool for geometry homework, design patterns, craft templates, architecture sketches, and regular polygon measurement checks.
- Choose the input you know first.
- Check whether the pentagon is regular or irregular.
- Do not round intermediate values too early.
Related calculators
Use these shape tools for related regular polygon and geometry questions.
Pentagon Calculator FAQ
How do I find pentagon perimeter?
For a regular pentagon, multiply the side length by 5.
How do I find the area of a regular pentagon?
Use area = perimeter x apothem / 2 when you know the apothem, or use a side-length formula for a regular pentagon.
What is the diagonal of a regular pentagon?
The diagonal equals side length multiplied by about 1.61803398875, the golden ratio.
Can this calculator work for irregular pentagons?
It is for regular pentagons. Irregular pentagons require different measurements or coordinate geometry.
What units should I use?
Use any length unit, but keep it consistent. Area will be in square units.
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