Exterior-angle sum
Add one consistently directed exterior angle at every vertex of a polygon.
Exterior-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 exterior-angle sum works
Walk once around the outside of the polygon. At each corner you turn through that corner's exterior angle, and by the time you arrive back at your starting point facing your original direction, you have turned through exactly one complete revolution.
What each symbol means
Each exterior angle is the turning angle at a vertex.
Exterior-angle sum: when it holds
Use one exterior angle per vertex and traverse the polygon in one direction.
When it stops applying
Every corner may contribute exactly one turn, chosen consistently as you travel in a single direction. Taking both exterior angles at some corners, or reversing direction partway around, destroys the single-revolution argument.
Exterior-angle sum: a worked example
A regular octagon has each exterior angle $360^\circ/8=45^\circ$.
The mistake to avoid
What people do: Expecting the total to grow as the polygon gains more sides.
Why it goes wrong: The total stays at 360 degrees for every convex polygon; only the individual turns get smaller as more corners share that one full revolution.
Do this instead: For a regular polygon, divide 360 by the number of sides rather than multiplying anything by it.
Exterior-angle sum: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. Each exterior angle is the turning angle at a vertex.
- Check the conditions before substituting. Use one exterior angle per vertex and traverse the polygon in one direction.
- 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 exterior-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 exterior-angle sum
Do all the turns have to be the same size?
No. They add to a full revolution for any convex polygon, and only in a regular one does every turn happen to be equal to the others.
How do I find the number of sides from one exterior angle?
Divide 360 by that angle. An exterior angle of 24 degrees means the regular polygon has 15 sides.
How does an exterior angle relate to the interior one beside it?
They sit on a straight line and so add to 180 degrees, which means an interior angle of 135 degrees pairs with an exterior angle of 45.
Why does the total ignore the number of sides?
Because it measures one complete trip around the shape, and a full revolution is 360 degrees no matter how many corners you turned at along the way.