Pentagon Area Calculator

A regular pentagon is a five-sided polygon with equal-length sides and identical interior angles, each measuring exactly 108 degrees. But it’s more than just a neat shape: regular pentagons show up in nature, art, architecture, and even national defense buildings. So, how do you calculate the area of one? And when would you need to?

Want to learn more about other shapes? Try our full collection of Area Calculators.

What Is a Regular Pentagon?

A regular pentagon is a five-sided polygon where all sides are of equal length and all internal angles are the same—exactly 108 degrees each. The word "pentagon" comes from the Greek pente (five) and gonia (angle), which tells you almost everything you need to know.

You’ve probably seen a regular pentagon more often than you realize. The classic example? A stop sign—with its crisp, symmetrical edges—is often mistaken for an octagon, but regular pentagons show up in more subtle ways. Look at the cross-section of okra, starfish patterns, certain flower petals, or even in Islamic tile art. Their balanced, compact shape makes them visually pleasing and structurally practical.

In geometry, regular polygons like this are often used as building blocks in problem-solving because of their symmetry. A regular pentagon is especially interesting because it cannot tile a flat surface perfectly on its own (unlike squares or equilateral triangles), which gives it a special place in mathematical puzzles and design challenges.

If you're wondering whether the Pentagon—the U.S. Department of Defense headquarters—is actually shaped like a regular pentagon, not quite. It’s pentagonal, yes, but not regular. Still, its iconic five-sided design draws direct inspiration from the regular pentagon, and the name sticks for a reason.

Try out Regular Polygon Area Calculator

How to Calculate the Area of a Regular Pentagon

Wondering how to determine the area of a regular pentagon? Fortunately, there’s a well-defined formula—though at first glance, it might seem a bit advanced.

Method 1: The Trigonometric Formula

To compute the area using trigonometry, take five-fourths (5 ÷ 4), multiply it by the square of the side length, and then multiply that by the cotangent of π divided by five.

In clearer terms:

Area = (5/4) × s² × cot(π/5)

Where:

• s = length of a side

• π/5 = angle in radians

• cot(π/5) ≈ 1.37638

If you’re not familiar with trigonometric terms, don’t worry—the cotangent of π/5 is simply a constant, approximately 1.37638. So, once you know the side length, you can plug it into the formula for a quick solution, especially if you're using a calculator or an online pentagon area tool.

Method 2: Apothem Formula (Simpler for Practical Use)

For most practical uses—like school assignments or DIY projects—this method is usually easier to follow. Multiply the pentagon’s perimeter by the apothem (a line drawn from the center to the midpoint of one side), then divide the result by two.

Simplified version:

Area = (Perimeter × Apothem) ÷ 2

• Multiply the side length by 5 to get the perimeter

• Multiply that by the apothem (a line from the center to the midpoint of a side)

• Divide by 2 for the final area

This formula is especially useful when you can measure both side lengths and the apothem directly.

Why Are Pentagon Angles 108 Degrees?

Every regular pentagon has five equal internal angles, and each of them measures exactly 108 degrees. But why 108? And is that number just a geometric coincidence, or something more?

There’s a simple formula to find any polygon's interior angle:

(n - 2) × 180 ÷ n

For a pentagon:

• (5 - 2) × 180 = 540° total

• 540 ÷ 5 = 108° per angle

What’s fascinating is how often that number appears—not just in math, but in nature and culture.

Pentagons in Nature

Take a closer look around: many flowers, such as morning glories and wild roses, show fivefold radial symmetry, with shapes that mimic a pentagon. The starfish is another classic example, with five arms spaced roughly 108 degrees apart.

In molecular chemistry, five-sided ring structures appear in carbon-based compounds, contributing to their stability. And yes—even slicing an apple or okra crosswise may reveal a subtle pentagonal shape in the core.

Try out Rhombus Area Calculator

The Spiritual Significance of 108

While 108 is clearly important in geometry, it also holds deep spiritual meaning in many cultures. In Hinduism, Buddhism, and yoga traditions, 108 is considered sacred. You’ll find it in prayer beads, astrological calculations, and meditative practices.

Is this just a coincidence? Maybe. But it’s intriguing how this geometric constant connects with philosophy, spirituality, and science, all through the shape of a regular pentagon.