Exterior Angles of a Polygon

Drive once around a city block and you make a series of turns; by the time you are back where you started, you are facing the same way you set off — you have turned through one full circle. Walk the edge of any polygon-shaped field and the same thing happens, and that single idea is what this page is about.

Here is a fact that never stops being delightful. Take any convex polygon — a triangle, a hexagon, a fifty-sided shape, a shape with a hundred sides — and measure the little turn you make at each corner as you walk around it. Add all those turns together and you always, always, get exactly 360^\circ.

Not "about" 360. Not "360 for some shapes." Every convex polygon, every time, no matter how many sides. The interior angles grow and grow as you add sides — but the turns around the outside stubbornly refuse to change. That surprising steadiness is what this page is about.

\text{sum of exterior angles} = 360^\circ

The exterior angle at a corner is the angle between one side and the extension of the side before it — the amount you swing your body through as you round that corner. It builds straight on the angle sum of a polygon and on the exterior angle of a triangle: at every corner, the interior angle and its exterior angle sit on a straight line together.

Walk once around a convex polygon, turning at each corner. Then:

Why it works: take a walk

No formula needed — just go for a walk. Imagine strolling along the edge of the polygon like a path around a field. At each corner you swing round to face along the next side; that swing is the exterior angle. Keep going until you are back where you started, facing the original direction. You have spun through one complete turn — one whole circle, 360^\circ. Step through it.

The five turns return you to your starting direction, so together they make a complete 360^\circ. The same walk works for a triangle, a hexagon, or any convex polygon — you always end up having turned exactly once around, so the total is always 360^\circ. Adding sides just means more corners, each with a smaller turn; the turns get tinier but their total holds firm.

Two sums, two very different lives

It is worth seeing the contrast on one picture. As a polygon gains sides, the interior angle sum climbs a steady staircase — but the exterior angle sum lies perfectly flat at 360^\circ forever.

Worked examples

1. Each exterior angle of a regular octagon. An octagon has n = 8 equal corners sharing the full 360^\circ of turning:

\text{each exterior angle} = \frac{360^\circ}{8} = 45^\circ.

2. Working backward from a turn. A regular polygon has each exterior angle equal to 24^\circ. How many sides? The turns must total 360^\circ, so count how many 24^\circ turns fit:

n = \frac{360^\circ}{24^\circ} = 15 \text{ sides}.

3. Switching between interior and exterior. They always add to 180^\circ, so knowing one hands you the other. For a regular decagon (n = 10), the exterior angle is 360 \div 10 = 36^\circ, so each interior angle is:

180^\circ - 36^\circ = 144^\circ.

That is a far quicker route to a regular polygon's interior angle than splitting it into triangles!

Two traps hide here. First, the "exterior angles add to 360^\circ" rule is for convex polygons — shapes with no dents. If a polygon caves inward at a corner (a "reflex" dent), the neat walk breaks and the tidy total can fail.

Second — and this is the big one — do not muddle the two sums. The exterior-angle sum is always 360^\circ, frozen forever. The interior-angle sum is (n-2)\times 180^\circ and grows with every side. They are linked at each corner, where interior and exterior are supplementary — they sit on a straight line and add to 180^\circ — but their totals are completely different beasts. Reach for 360^\circ only when the question says exterior.

Program a Logo turtle — or a little robot vacuum — to drive along and turn at each corner, and the way it knows it has traced a closed loop is that its turns have added up to a full 360^\circ. That is the exterior-angle rule doing an honest day's work inside a machine: total turn once around a simple shape = one full circle.

And the idea reaches far higher than school. Mathematicians call the total amount of turning around a curve its total curvature, and there is a beautiful theorem that says the total curvature of any simple closed curve — a wobbly blob, a circle, a polygon — is exactly 360^\circ. Your tidy pentagon fact is the first step onto a staircase that climbs all the way up to the geometry of curved surfaces and the shape of space itself.

See it explained