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 actually need to?

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

The Formula to Calculate the Area of a Regular Pentagon

So how do you actually find the area of a regular pentagon? Thankfully, there’s a clear formula for that—though it might look a bit technical at first glance.

Option 1: The Trigonometric Formula

To calculate the area, take five-fourths (or 5 divided by 4), multiply it by the square of the side length, and then multiply that by the cotangent of pi divided by five.

In simple terms:

Area=(54)×(side length)2×cot⁡(π5)If you're not into trigonometry, don't worry—the cotangent of π/5 is just a fixed number, approximately 1.37638. So if you know the side length, you can plug it into this formula and get the area quickly (especially with a calculator or an online tool).

Option 2: The Apothem-Based Formula (Simpler for Most People)

If you're working on a school problem or a hands-on project, this formula is often more intuitive. Multiply the perimeter of the pentagon by the apothem (a line from the center to the middle of one side), then divide the result by two.

Put simply: Area = (Perimeter × Apothem) ÷ 2

To find the perimeter, just multiply the side length by five. Then multiply that by the apothem, and divide the result by two. That’s your area.

closed within its five sides. For irregular pentagons, the process of finding the area can be complex, but for regular pentagons, there's a straightforward geometric formula.

Why 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?

Let’s start with the math. For any polygon, the formula to find the size of an internal angle is:

(n − 2) × 180, divided by n,
where n is the number of sides.

For a pentagon, that becomes:
(5 − 2) × 180 = 540 degrees in total
Then divide by 5 sides:
540 ÷ 5 = 108 degrees per angle

That number—108—pops up in more places than just geometry class.

Nature Loves a Pentagon Too

Take a closer look at the world around you. Many flowers, such as morning glories or wild roses, have fivefold radial symmetry that hints at a pentagonal structure. Sea creatures like starfish follow a similar pattern—five arms radiating from a central point, each separated by angles close to 108 degrees.

In molecular chemistry, certain carbon-based structures also adopt five-sided patterns for their stability. And even the cross-section of an okra or an apple sliced the right way can reveal a faint pentagon hiding in plain sight.

Try out Rhombus Area Calculator

And Then There’s the Spiritual Side of 108

This one’s more cultural than scientific, but still fascinating: in many Eastern philosophies, the number 108 is considered sacred. It appears in Buddhist malas (prayer beads), Vedic astrology, and yoga traditions. Coincidence? Probably. But it adds to the sense that 108 has a special symmetry, even beyond math.