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Pentagon Area Calculator

Calculate the area of a regular pentagon with five sides of equal length. Includes the formula, a worked example, and reference sizes for architecture.

Calculate area of pentagon

Formula

Regular pentagon:

textarea=frac14sqrt5(5+2sqrt5)timess2\\text{area} = \\\\frac{1}{4} \\\\sqrt{5(5 + 2\\\\sqrt{5})} \\\\times s^2

Where s is the side length.

Approximation:

textareaapprox1.7205timess2\\text{area} \\\\approx 1.7205 \\\\times s^2

Worked example

Calculate the area of a regular pentagon with side length 6 cm:

textarea=frac14sqrt5(5+2sqrt5)times62\\text{area} = \\\\frac{1}{4} \\\\sqrt{5(5 + 2\\\\sqrt{5})} \\\\times 6^2=frac14sqrt5(5+4.472)times36= \\\\frac{1}{4} \\\\sqrt{5(5 + 4.472)} \\\\times 36=frac14sqrt42.36times36= \\\\frac{1}{4} \\\\sqrt{42.36} \\\\times 36=frac14times6.509times36=58.58textcm2= \\\\frac{1}{4} \\\\times 6.509 \\\\times 36 = 58.58 \\\\text{ cm}^2

Result: 58.58 cm²

How to use

  1. Enter the side length of the regular pentagon
  2. Click "Calculate" to see the area

Related formulas

  • Perimeter: 5timestextside5 \\\\times \\\\text{side}
  • Apothem: a=fracs2tan(36°)a = \\\\frac{s}{2 \\\\tan(36°)}
  • Area with apothem: frac12timestextperimetertimestextapothem\\\\frac{1}{2} \\\\times \\\\text{perimeter} \\\\times \\\\text{apothem}

Common use cases

  • Architecture (pentagonal structures)
  • Soccer ball patterns
  • Company logos
  • Art and design

FAQ

What if the pentagon is irregular? You need to divide it into triangles or use other methods.

What is the sum of interior angles? 540° for any pentagon.

Can I calculate from the radius? Yes, use s=2Rsin(36°)s = 2R \\\\sin(36°) to find the side first.

How many diagonals does a pentagon have? 5 diagonals.

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