Understand the regular pentagon

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.

Properties that make the calculations work

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.

Where the model is useful

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.

Measure the right thing

The apothem reaches a side midpoint; the circumradius reaches a vertex. They are different lengths.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k². This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Regular pentagon provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.