Pentagon Calculator

Category: Geometry
- December 26, 2025
| |

Calculate and visualise properties of regular pentagons. A regular pentagon has five equal sides and angles. This calculator helps you find various measurements including area, perimeter, diagonals, and more.

Pentagon Calculator

units

Display Options

About Regular Pentagons

A regular pentagon is a five-sided polygon with equal sides and equal interior angles. It's one of the basic regular polygons studied in geometry, with a rich set of properties.

Key Properties of Regular Pentagons
  • Sides: 5 equal sides
  • Interior Angles: 5 equal angles of 108° each
  • Exterior Angles: 5 equal angles of 72° each
  • Diagonals: 5 diagonals
  • Symmetry: 5 lines of symmetry and rotational symmetry of order 5
Formulas for Regular Pentagons

Where a is the side length:

  • Perimeter: P = 5a
  • Area: A = (5/4) × a² × cot(π/5) ≈ 1.72 × a²
  • Diagonal: d = a × (1 + √5)/2 ≈ 1.618 × a (the golden ratio)
  • Inradius: r = a × cot(π/5) / 2 ≈ 0.688 × a
  • Circumradius: R = a / (2 × sin(π/5)) ≈ 0.85 × a
The Golden Ratio in Pentagons

The regular pentagon is closely related to the golden ratio (φ ≈ 1.618). The ratio of a diagonal to a side is exactly the golden ratio. The golden ratio appears in many natural, artistic, and architectural designs.

Applications of Regular Pentagons

Regular pentagons appear in many contexts:

  • Architecture and design
  • Military fortifications (pentagonal forts)
  • The regular dodecahedron (a Platonic solid with pentagonal faces)
  • The shape of many flowers and fruits in nature
  • Corporate logos and design elements
  • The Pentagon building (headquarters of the U.S. Department of Defense)

What Is the Pentagon Calculator?

The Pentagon Calculator is a handy tool designed to help you calculate different properties of regular pentagons. A regular pentagon has five equal sides and angles, making it a fascinating shape in Geometry. With this calculator, you can find the area, perimeter, the length of diagonals, and more. It’s perfect for students, teachers, or anyone interested in geometry and Mathematics.

Understanding the Properties of Pentagons

Regular pentagons have unique properties that make them interesting. Here are some key features:

  • Sides: The pentagon has 5 equal sides.
  • Interior Angles: Each angle measures 108°.
  • Exterior Angles: Each angle measures 72°.
  • Diagonals: A regular pentagon has 5 diagonals.
  • Symmetry: It has 5 lines of symmetry.

Calculating Area and Perimeter

The calculator lets you find both the area and perimeter of a pentagon easily. If you know the length of one side, you can quickly determine the perimeter by multiplying the side length by five. The area, on the Other hand, requires a bit more calculation. Using formulas, the calculator does the heavy lifting for you!

Finding Diagonal Lengths

Diagonals in a pentagon connect non-adjacent vertices. The Pentagon Calculator can calculate the length of a diagonal based on the side length. Knowing diagonal lengths is useful in various applications, including design and geometry projects.

Inradius and Circumradius Information

The inradius is the radius of the inscribed circle, while the circumradius is the radius of the circumscribed circle around the pentagon. The calculator provides these values, which are important for understanding the relationship between the pentagon's size and its circular properties. Here’s what you can learn:

  • Inradius: Measures how far the inscribed circle reaches inside the pentagon.
  • Circumradius: Measures how far the circumscribed circle reaches outside the pentagon.

Displaying Results and Calculation Steps

After entering your values, the calculator displays the results in a clear format. You can see the side length, perimeter, area, diagonal lengths, inradius, and circumradius all in one place. Additionally, if you choose, it can show you the calculation steps, helping you learn and understand the maths behind the results.

Visualising Your Pentagon

The Pentagon Calculator includes a visual representation of the pentagon based on the inputs you provide. This can enhance your understanding of the shape and its properties. Seeing the pentagon drawn out helps reinforce the concepts you've calculated and can make learning more engaging.

Applications of the Pentagon Shape

Pentagons are not just theoretical shapes; they appear in real life too. Regular pentagons can be found in architecture, design, and nature. Here are a few places where you might see them:

  • Design elements in logos and branding.
  • Architectural structures, like the Pentagon building.
  • Patterns found in flowers and fruits.
  • Military fortifications.