GeometryFree

Pentagon Area Calculator

Pentagon area calculator finds the area of a regular pentagon from its side length. A pentagon with side 5 has area approximately 43.01 square units. Uses the formula A = (s^2 x sqrt(25 + 10sqrt(5))) / 4.

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Side Length (s)

Result

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Formula

Core Formula

A = \frac{s^2}{4} \sqrt{25 + 10\sqrt{5}}

How it works: For a regular pentagon, the area simplifies to approximately 1.7205 x s^2.

Worked Example

Find the area of a regular pentagon with side 5.

1Step 1: A = (s^2 / 4) x sqrt(25 + 10sqrt(5)).

2Step 2: sqrt(5) = 2.2361, so 10 x 2.2361 = 22.361.

3Step 3: 25 + 22.361 = 47.361, sqrt(47.361) = 6.882.

4Step 4: A = (25 / 4) x 6.882 = 6.25 x 6.882 = 43.01 sq units. The apothem = s / (2 x tan(36)) = 5 / (2 x 0.7265) = 3.44. Perimeter = 5 x 5 = 25.

How to Calculate Regular Pentagon Area

A regular pentagon has five equal sides and five equal interior angles of 108 degrees each. It has deep connections to the golden ratio.

Area formula: A = (s^2/4) x sqrt(25+10sqrt(5)), approximately 1.7205 x s^2
Perimeter = 5 x side length. A pentagon with side 5 has perimeter 25
Apothem (center to midpoint of side) = s / (2 x tan(36)), approximately 0.6882 x s
The diagonal of a regular pentagon divided by its side equals the golden ratio phi = 1.618

Regular pentagons appear in architecture, nature (starfish, okra cross-sections), and mathematical tiling patterns.

You can also calculate changes using our Hexagon Area Calculator, Octagon Area Calculator or Circle Geometry Theorem Solver.

Frequently Asked Questions

What is the area of a regular pentagon with side 5?

A = (25/4) x sqrt(25 + 10sqrt(5)) = 6.25 x 6.882 = 43.01 square units. The perimeter is 25 and the apothem is approximately 3.44.

What is the interior angle of a regular pentagon?

Each interior angle is 108 degrees. The sum of all interior angles is (5-2) x 180 = 540 degrees. Each exterior angle is 72 degrees (360/5).

How is the pentagon related to the golden ratio?

The diagonal of a regular pentagon divided by its side length equals the golden ratio phi = (1+sqrt(5))/2 = 1.618. This ratio appears throughout the pentagon: the diagonals form a smaller pentagon inside, creating an infinite nested pattern.

Is this calculator only for regular pentagons?

Yes. This formula works only for regular pentagons where all five sides are equal and all five angles are 108 degrees. For irregular pentagons, you need to divide the shape into triangles and sum their areas.

How does pentagon area compare to hexagon area for the same side?

For the same side length, a hexagon has more area. With side 5: pentagon area = 43.01, hexagon area = 64.95. The hexagon is about 51% larger because it has one more side and tiles more efficiently.

How do I add this Pentagon Area Calculator to my site?

Absolutely. Use the "Embed" option above to tailor the dimensions, color scheme, and styling to match your site. Copy the generated iframe snippet and drop it into your HTML, WordPress editor, or any CMS. There is no cost and no account required. See calculory.com/services/embed-calculators for a step-by-step walkthrough.

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I can help you understand the pentagon area calculator formula, interpret your results, and answer follow-up questions.

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