Regular polygon angles
Find equal interior and central angles of a regular polygon.
Regular polygon angles is one of 4 polygons formulas in the geometry section of this library, and it is used at middle school · high school level.
Why regular polygon angles works
In a regular polygon every interior angle is identical, so sharing the known total equally among the corners gives each one. The central angles come from a different sharing: a full turn at the centre is divided evenly among the identical triangles that meet there.
What each symbol means
$n$ is the number of sides.
Regular polygon angles: when it holds
$n\ge3$ and the polygon must be regular.
When it stops applying
Equal angles require a genuinely regular polygon. Lean a rectangle over into a parallelogram and it still has four sides and 360 degrees in total, but the angles come in two different sizes and dividing evenly means nothing.
Regular polygon angles: a worked example
A regular hexagon has interior angles $120^\circ$ and central angles $60^\circ$.
The mistake to avoid
What people do: Dividing 360 by the number of sides and calling the result an interior angle.
Why it goes wrong: That quotient is the central angle, which is also the exterior angle; for a hexagon it gives 60 degrees when the interior angle is 120.
Do this instead: Work out the total for the polygon first, then divide that total by the number of corners.
Regular polygon angles: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is the number of sides.
- Check the conditions before substituting. $n\ge3$ and the polygon must be regular.
- 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
- Middle school · 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 angles 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 angles
Why does the central angle match the exterior angle?
Both come from splitting one complete revolution equally among the corners, so both are 360 divided by the number of sides.
Which regular polygons tile a floor on their own?
Triangles, squares and hexagons, because their interior angles of 60, 90 and 120 degrees each divide exactly into 360.
What happens to the interior angle as sides are added?
It creeps upward toward 180 degrees without ever reaching it, as the outline flattens into something closer and closer to a circle.
What are the angles in a regular pentagon?
The interior angle is 540 divided by 5, which is 108 degrees, and the central angle is 360 divided by 5, which is 72 degrees.