Interior-angle sum is one of 4 polygons formulas in the geometry section of this library, and it is used at middle school · high school level.
Why interior-angle sum works
Choose one corner and draw every diagonal from it. Those diagonals cut the polygon into two fewer triangles than it has sides, none of them overlapping, and every triangle contributes 180 degrees, all of which lands inside the polygon.
What each symbol means
$n$ is the number of sides.
Interior-angle sum: when it holds
$n$ is an integer with $n\ge3$; the polygon is simple.
When it stops applying
Each corner may be counted only once, which requires a simple polygon whose edges do not cross. A five-pointed star has five corners but its point angles add to 180 degrees, nowhere near the 540 this rule would predict.
Interior-angle sum: a worked example
A hexagon has sum $(6-2)180^\circ=720^\circ$.
The mistake to avoid
What people do: Multiplying the number of sides by 180 with no subtraction.
Why it goes wrong: That counts two triangles too many, so a hexagon is reported at 1080 degrees instead of its true 720.
Do this instead: Subtract 2 from the number of sides before you multiply by 180.
Interior-angle sum: 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. $n$ is an integer with $n\ge3$; the polygon is simple.
- 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 interior-angle sum 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 interior-angle sum
Does the polygon have to be regular?
No. Any simple polygon with the same number of sides has the same total; only the way that total is shared between the corners changes.
What is the total for a four-sided figure?
It is 360 degrees, since two triangles are needed to fill it, which is why the angles of any quadrilateral add to a full turn.
Can I use this to find one missing angle?
Yes. Add up the angles you know and subtract that from the total. In a pentagon totalling 540, four known angles determine the fifth.
What happens for a polygon with very many sides?
The total climbs by another 180 degrees for each extra side, so a 100-sided polygon reaches 17640 degrees.