Regular polygon area
Find area from apothem and perimeter or side count and side length.
Regular polygon area is one of 4 polygons formulas in the geometry section of this library, and it is used at high school level.
Why regular polygon area works
Join the centre to every corner and the polygon splits into identical triangles, one per side. Each has a side of the polygon as its base and the apothem as its height, so the total is half the apothem times all the bases put together, and all the bases together are the perimeter.
What each symbol means
$a$ is apothem, $P$ perimeter, $n$ side count, and $s$ side length.
Regular polygon area: when it holds
The polygon must be regular and $n\ge3$.
When it stops applying
Both versions assume the polygon is regular, with every side and every angle equal. An irregular pentagon has no single apothem at all, so this gives a wrong answer and the shape must be cut into triangles that are measured one by one.
Regular polygon area: a worked example
A regular hexagon with side $2$ has $A=6(4)/(4\tan(\pi/6))=6\sqrt3$.
The mistake to avoid
What people do: Using the distance from the centre out to a corner as the apothem.
Why it goes wrong: The apothem reaches only the midpoint of a side, which is shorter: on a hexagon of side 2 it is about 1.732 while the distance to a corner is 2.
Do this instead: Measure the perpendicular from the centre to a side, not the slanted line from the centre to a corner.
Regular polygon area: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is apothem, $P$ perimeter, $n$ side count, and $s$ side length.
- Check the conditions before substituting. The polygon must be regular and $n\ge3$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Polygons
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where regular polygon area comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about regular polygon area
What exactly is the apothem?
It is the perpendicular distance from the centre to the middle of any side. In a regular polygon that same distance works for every side.
Why does the tangent version assume radians?
The angle inside the tangent is written as π divided by the number of sides, which is radian measure, so a calculator in degree mode needs 180 divided by the number of sides.
What happens as the number of sides grows?
The polygon closes in on a circle, and the value approaches π times the apothem squared, with the apothem taking the role of the radius.
What is the area of a regular hexagon with side 2?
It is 6 times the square root of 3, about 10.39 square units, which is the same as six equilateral triangles of side 2 placed together.