Pentagon Calculator
Area and perimeter of a regular pentagon.
Area
61.9372
regular pentagon
Perimeter
30
Interior angle
108°
AI Breakdown & Smart Takeaway
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How the Pentagon Calculator works
The Pentagon Calculator instantly computes the area and perimeter of any regular pentagon from a single side length, saving you from manual trigonometric calculations. It's ideal for students, architects, engineers, designers, and hobbyists working with five-sided symmetrical shapes in geometry, construction, or creative projects.
A regular pentagon is a five-sided polygon where every side is equal in length and every interior angle measures exactly 108 degrees. Because all sides are identical, you only need one measurement — the side length — to fully define the shape and calculate every geometric property, including area and perimeter. This elegant property of regular polygons is what makes the calculator so efficient: enter one number and instantly get both key measurements with full precision.
The perimeter is the simpler of the two calculations — it is simply five times the side length (P = 5s). The area formula is more involved, derived from dividing the pentagon into five congruent isosceles triangles that all meet at the center. The area formula uses the tangent of 54 degrees (which comes from the interior geometry of those triangles), yielding A = (5/4) × s² × tan(54°). Since tan(54°) is an irrational constant (approximately 1.37638), a calculator handles this far more reliably than manual arithmetic, especially when precision matters for real-world applications.
A common mistake is confusing a regular pentagon with an irregular pentagon. This calculator is specifically designed for regular pentagons — shapes where all sides and all angles are equal. If your five-sided figure has sides of different lengths or unequal angles, the formulas used here will not give you the correct area or perimeter. Another frequent error is mixing up units: if your side length is in centimeters, your area result will be in square centimeters, not square meters. Always confirm your input units before interpreting the output.
In practical applications, understanding the apothem — the perpendicular distance from the center of the pentagon to the midpoint of any side — is often just as important as the area itself. The apothem is embedded in the area formula and equals (s/2) × tan(54°). Architects and tile designers frequently use the apothem when fitting pentagons into a layout or calculating the radius of an inscribed circle. If you need the apothem or circumradius for a specific project, those values can be derived directly from the side length using straightforward trigonometry once you understand the underlying geometry.
Formula
Area = ¼·√(5(5+2√5))·s²
Pro tips
- Always double-check your unit of measurement before entering the side length — area scales as the square of the unit, so a side in meters gives area in square meters, not square centimeters.
- If you know the perimeter rather than the side length, simply divide by 5 to get the side length, then use that value as your input.
- For tiling or flooring projects, calculate the area of one regular pentagon tile, then divide the total floor area by that value to estimate the number of tiles needed — add 10–15% for cuts and waste.
- When comparing a pentagon to other regular polygons of the same perimeter, note that the pentagon encloses more area than a square or equilateral triangle with the same perimeter, illustrating why hexagonal and pentagonal shapes appear efficiently in nature.
- If you need the apothem for a physical project (such as cutting a centered hole or fitting an inscribed circle), compute it as approximately 0.6882 × s, which is a quick mental shortcut derived from (s/2) × tan(54°).
Key terms
- Regular Pentagon
- — A five-sided polygon in which all five sides are equal in length and all five interior angles each measure exactly 108 degrees.
- Side Length (s)
- — The length of one edge of the regular pentagon, which is the sole input needed to calculate all other geometric properties of the shape.
- Apothem
- — The perpendicular distance from the center of a regular polygon to the midpoint of one of its sides, equal to (s/2) × tan(54°) for a regular pentagon.
- Perimeter
- — The total distance around the outside of the pentagon, calculated as five times the side length.
- Circumradius
- — The radius of the circle that passes through all five vertices of the pentagon, equal to s / (2 × sin(36°)).
- Interior Angle
- — The angle formed inside the pentagon at each vertex; for a regular pentagon, this is always exactly 108 degrees.
