Look a little closer.

Five equal sides and a golden connection. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Its diagonal-to-side ratio is the golden ratio.

Each interior angle is 108 degrees and each central angle is 72 degrees. Joining the centre to the vertices creates five congruent isosceles triangles. Their shared perpendicular height is the apothem.

02 / A different perspective

Its diagonals form a pentagram.

A regular pentagon has five equal sides and five equal interior angles. An arbitrary five-sided polygon does not satisfy the same side-only formulas. All vertices lie on a common circumcircle.

03 / A connection to explore

A regular pentagon alone cannot tile a plane edge-to-edge without gaps.

Choose a side, circumradius, apothem or target area to size a regular pentagon. The calculators assume perfect regularity and do not infer a general pentagon from a single measurement.

Try the idea with numbers

Open the from a side calculator, note its outputs, then multiply each length input by two. Area should increase by four and every reported length should double, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Regular pentagon at Wolfram MathWorld.