Before using a polygon formula, decide what the problem actually gives. Counting six sides tells you the total interior angle sum. It does not tell you that each angle is 120°. That extra step needs equal-angle information. A tidy drawing is not evidence of regularity.
All polygons in this lesson are convex: their sides do not cross and every interior angle is less than 180°. We use one consistent exterior turn at each vertex, obtained by continuing the incoming side forwards and turning to the outgoing side.
Build the angle-sum rule rather than guessing it
Choose one vertex of a convex polygon with n sides. Draw diagonals from it to the non-adjacent vertices. The diagonals stay inside the polygon, producing n − 2 triangles without overlap. Each triangle contributes 180°, so the polygon’s interior-angle sum is (n − 2) × 180°.
This is a total, not a statement about equal shares. A triangle has one triangle’s worth, 180°; a quadrilateral has two, 360°; a pentagon has three, 540°. Adding one side increases the total by 180°, but it need not increase every individual angle.
Case 1: an irregular hexagon
A convex hexagon ABCDEF has consecutive interior angles 120°, 95°, 140°, 130°, 110° and z. Find z. The drawing below has deliberately unequal angles; no regularity is given or needed.
There are six sides, so the total is (6 − 2) × 180° = 720°. The five known angles add to 120° + 95° + 140° + 130° + 110° = 595°. Therefore z = 720° − 595° = 125°.
Check the complete sum: 120° + 95° + 140° + 130° + 110° + 125° = 720°. Each angle is between 0° and 180°, consistent with the stated convex figure. That check does not prove every arbitrary list is geometrically realizable, but it catches a negative or reflex result that would contradict this problem.
| Vertex | Interior angle | Exterior turn: 180° minus interior |
|---|---|---|
| A | 120° | 60° |
| B | 95° | 85° |
| C | 140° | 40° |
| D | 130° | 50° |
| E | 110° | 70° |
| F | 125° | 55° |
| Total | 720° | 360° |
Walking around this hexagon, you turn in the same sense at each corner. One full circuit restores the starting direction, so those exterior turns total one full turn, 360°. They are unequal, but their total is fixed.
Case 2: recover the side count of a regular polygon
A polygon is explicitly stated to be regular, and one exterior turn is 24°. Find its number of sides and each interior angle. Regular means both equal sides and equal angles, so all exterior turns are equal too.
If n equal turns complete 360°, then n × 24° = 360° and n = 15. The adjacent interior angle and exterior turn form a straight angle, so each interior angle is 180° − 24° = 156°.
Check by the other formula: the total for 15 sides is (15 − 2) × 180° = 2340°. Dividing by 15 gives 156°. Conversely, 15 × 156° = 2340° and 15 × 24° = 360°. Both routes agree, and 15 is an integer at least three.
Keep the ray choice visible
At a vertex there are several angles you could draw outside a polygon. The exterior turn used here is the supplement of the interior angle: together they make 180°, not 360°. Extending the wrong ray or measuring the reflex angle can make a correct formula produce the wrong value. Trace the incoming direction and then the outgoing direction before marking the turn.
Case 3: reject an impossible exact claim
A caption claims that a regular convex polygon has every interior angle exactly 151°. Is this possible? Its exterior turn would be 180° − 151° = 29°, so n = 360° ÷ 29° = 360/29, approximately 12.414. There cannot be a fraction of a side in a polygon. No regular polygon satisfies that exact claim.
Rounding n to 12 changes the problem. A regular 12-sided polygon has exterior turns 30° and interior angles 150°, not 151°. A regular 13-sided polygon has interior angles 180° − 360°/13, approximately 152.308°. Neither agrees with the exact value.
If an angle were measured approximately, the wording would need a tolerance, such as the nearest degree. You would then test whether an integer side count produces an angle in that permitted rounding interval. Do not silently convert exact data into approximate data to rescue a contradiction.
Separate equal sides, equal angles and regularity
A rectangle has four equal angles but need not have four equal sides. It is equiangular, so its angle sum may be divided equally, but it is regular only when it is a square. A rhombus has four equal sides but need not have equal angles. Therefore “equilateral” alone does not license the equal-angle formula.
Likewise, the irregular hexagon in Case 1 has average angle 720° ÷ 6 = 120°, yet only one of its angles is actually 120°. An average describes the total spread over equal shares; it does not replace the supplied unequal values.
Three solved practice tasks
- A convex pentagon has interior angles 88°, 112°, 125°, 105° and t. Find t. Its total is 3 × 180° = 540°. The known total is 430°, giving t = 110°. The exterior turns are 92°, 68°, 55°, 75° and 70°, which total 360°.
- A regular polygon has each interior angle 162°. Find n. The exterior turn is 180° − 162° = 18°. Thus n = 360° ÷ 18° = 20. Its interior total is 18 × 180° = 3240°, equal to 20 × 162°.
- Can a regular polygon have an exact exterior turn of 28°? The implied side count is 360/28 = 90/7, not an integer. No such exact regular polygon exists. Rounding to 13 would instead require an exterior turn of 360°/13, approximately 27.692°.
A final three-question check
Ask: How many sides are there? Which equalities are actually stated? Which angle is marked? Use the interior sum for a total, the 360° turn sum for a full circuit, and division into equal angles only when justified. Then check that a recovered side count is a whole number and that every answer agrees with the original data.
Frequently asked questions
Does a regular polygon only need equal sides?
No. It must have equal sides and equal interior angles. Equal sides alone do not justify using one common interior angle; a non-square rhombus is a counterexample.
Can I divide the interior-angle sum by n for an irregular polygon?
That division gives the average interior angle. It gives every interior angle only when all the angles are known to be equal.
Does the 360° exterior-turn sum require regularity?
No. One consistently directed exterior turn at each vertex of any convex polygon totals 360°. Regularity is needed to make all those turns equal.
Why are there n − 2 triangles?
In a convex n-sided polygon, draw diagonals from one vertex to every non-neighbouring vertex. The resulting non-overlapping triangles fill the polygon, and there are n − 2 of them.
Can I round the number of sides?
Not when the given angle is exact. A non-integer side count means no exact regular polygon fits the data. Rounded angle measurements require a stated tolerance before possible integers can be assessed.
Are all angles drawn outside a polygon exterior turns?
No. This lesson uses the smaller turning angle between a side continued forwards and the next side. A different outside angle might be reflex or use the wrong ray.
Does this method apply to star drawings?
The examples here are convex polygons with no crossing sides. Crossing star figures need a different treatment of paths and angles; do not transfer these counting steps without checking the figure.
Further reading
For the underlying mathematics, see OpenStax’s polygon definitions and angle relationships. The mathematics topic-test collection is a separate resource page that currently shows a member login; specific files and their answers are not verified here.
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