THE NUMORIX GUIDE
How to use the Pentagon Calculator
Last reviewed September 14, 2026
What this calculator does
For a regular pentagon with side s, perimeter = 5s, apothem = s/(2*tan(pi/5)), area = one-half * perimeter * apothem, and each interior angle = 108 degrees.
Formula and method
For a regular pentagon with side s, perimeter = 5s, apothem = s/(2*tan(pi/5)), area = one-half * perimeter * apothem, and each interior angle = 108 degrees.
Variables and inputs
Side Length defaults to 5 and represents one side of a regular pentagon. The regularity assumption is essential to the apothem formula.
Worked example
For s=5: perimeter = 25; apothem = 5/(2*tan(36 degrees)) = 3.4410; area = 0.5*25*3.4410 = 43.0119 square units; interior angle = 108 degrees.
How to interpret the result
The apothem is the perpendicular distance from the center to a side. For a regular polygon, area is one-half perimeter times apothem.
Common mistakes to avoid
Do not use an irregular pentagon with the regular-polygon formula, and do not confuse apothem with circumradius.
Assumptions and limitations
Only regular, convex pentagons are modeled. The engine assumes the side is positive and uses radians internally for tangent.
Practical use and checks
This calculator models a regular pentagon, not every five-sided outline. Enter side length 5 as a check; perimeter should be 25, apothem about 3.4410, area about 43.0119 square units, and each interior angle 108 degrees. The area result comes from one-half times perimeter times apothem. To check the geometry, imagine five congruent triangles from the center; their five central angles total 360 degrees and each interior angle is 108 degrees. Use the tool for a regular sign, tile, or polygon exercise when all sides and angles are intended to match. The apothem is the perpendicular distance from the center to a side, so it is not the same as the distance from the center to a vertex. Keep side and area units distinct, and do not add a waste percentage to the geometric area unless a material plan calls for it. An irregular or concave pentagon needs its vertices, diagonals, or a decomposition into simpler shapes; this single-side input cannot represent those cases. Very small or very large sides can make rounded trigonometric output look slightly inconsistent when you recompute it by hand. Use the regularity assumption as a decision check, not as a way to hide an irregular measured boundary.