Pentagon Calculator

Category: Geometry
- April 13, 2025
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Calculate and visualize 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

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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)

Regular Pentagon Calculator – Fast, Clear Geometry Help

The Regular Pentagon Calculator is a simple and effective tool to calculate and visualize the key properties of a regular pentagon. Whether you’re a student, teacher, or just someone curious about shapes, this calculator helps you understand how pentagons work by giving you instant results based on one known measurement.

Formulas Used in the Calculator

Perimeter: \( P = 5a \)

Area: \( A = \frac{5}{4} \times a^2 \times \cot\left(\frac{\pi}{5}\right) \approx 1.72 \times a^2 \)

Diagonal: \( d = a \times \frac{1 + \sqrt{5}}{2} \approx 1.618 \times a \)

Inradius (r): \( r = \frac{a}{2} \times \cot\left(\frac{\pi}{5}\right) \approx 0.688 \times a \)

Circumradius (R): \( R = \frac{a}{2 \times \sin\left(\frac{\pi}{5}\right)} \approx 0.85 \times a \)

What You Can Calculate

This tool lets you calculate all the following values for a regular pentagon, depending on which one you start with:

  • Side length
  • Perimeter
  • Area
  • Length of a diagonal
  • Inradius (radius of inscribed circle)
  • Circumradius (radius of circumscribed circle)
  • Interior and exterior angles

How to Use the Pentagon Calculator

Follow these quick steps to get accurate pentagon measurements:

  1. Select a value you already know (side, area, perimeter, etc.).
  2. Enter the number into the input box provided.
  3. Choose how many decimal places you'd like the results to show.
  4. Click the Calculate button.
  5. View the calculated properties, visual diagram, and optional step-by-step solution.

To start over or try a different value, click the Reset button.

Why Use This Calculator?

The Pentagon Calculator is a fast and friendly geometry helper for understanding how different pentagon measurements connect. Here’s how it helps:

  • Builds intuition about shape relationships using clear diagrams.
  • Helps check your math homework or geometry problems.
  • Visualizes results, making concepts more engaging and easier to grasp.
  • Includes exact formulas and optional detailed steps for learning.
  • Supports various input types, so you can work from whichever value you know.

It works well alongside other tools like the Area Calculator, Triangle Calculator, or Circle Calculator when comparing or working with multiple shapes.

Common Questions

Can I use this if I only know the diagonal?

Yes, just choose “Diagonal length” from the dropdown, enter the value, and the calculator will figure out everything else automatically.

What is the difference between inradius and circumradius?

The inradius is the distance from the center to a side (touches the inside), and the circumradius is the distance from the center to a vertex (covers the outside).

Can I see the math steps?

Yes. If you check “Show calculation steps,” you’ll get a breakdown of how each value is calculated using real formulas.

Do I need to understand trigonometry to use it?

No, the calculator handles all the formulas for you. It's a great support tool if you're just starting out with shapes or need a refresher.

Explore More Geometry Tools

This calculator pairs perfectly with other shape tools like:

Whether you're working on a math assignment, building something, or just curious about shapes, the Pentagon Calculator is a fast and clear way to calculate and understand pentagon geometry.