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This page sets out the method for angles in polygons in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Angles in polygons topic page.
When you know the number of sides and need the total or a missing interior angle.
When a question gives or asks for the number of sides of a regular polygon, the exterior angle is the quickest route.
One exterior angle is the turn at a corner if you walk around the shape.
For a regular polygon, each exterior angle is 360° ÷ n.
At each vertex, the interior and the exterior add to 180°.
They sit on a straight line.
A polygon with n sides splits into n − 2 triangles.
The interior angles add to (n − 2) × 180°.
In a regular polygon every side is equal and every interior angle is equal.
Divide the interior sum by n to get one interior angle.
If one exterior angle of a regular polygon is 30°, then n = 360 ÷ 30 = 12.
The exterior angle finds the number of sides.
Subtract the given interior angles from the interior sum.
Do not subtract them from 360° unless they are exterior angles.
Pick one vertex and draw every diagonal from it. An n-sided polygon splits into n - 2 triangles. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4. Each triangle has an angle sum of 180^°, so the polygon has interior sum (n - 2) × 180^°. That is why a pentagon sums to 3 × 180^° = 540^°, and why a hexagon sums to 4 × 180^° = 720^°.
This formula gives the sum, not one angle, unless the polygon is regular. If a pentagon is irregular, the five interior angles still add to 540^°, but they can be five different numbers. If it is regular, each interior angle is 540^° ÷ 5 = 108^°. Divide the sum only after you know the angles are equal.
Method: 1. Split the polygon into n-2 triangles. 2. Multiply by 180^°: the interior sum is (n-2)× 180^°. 3. A pentagon sums to 3× 180=540^°, and a hexagon sums to 4× 180=720^°.
Walk around the outside of a convex polygon and you finish facing the way you started. The turns you made are the exterior angles, and a full turn is 360^°. That sum is 360^° for a triangle, a pentagon, or a twenty-sided polygon. It does not grow with the number of sides. What changes is the size of each turn, if the polygon is regular.
For a regular polygon, divide 360^° by the number of sides. A regular hexagon has exterior angle 360^° ÷ 6 = 60^°. A regular octagon has exterior angle 360^° ÷ 8 = 45^°. This is the fast way to one exterior angle. Using (n - 2) × 180^° and then doing something else is the long way, and it is easy to forget you still have a sum rather than one angle.
Method: 1. Add the exterior angles of any convex polygon to 360^°. 2. One exterior angle of a regular polygon is 360^° divided by n. 3. A regular hexagon has exterior angle 360÷ 6=60^°.
At each vertex the interior angle and the exterior angle make a straight line, so they add to 180^°. Once you know one, subtract from 180^° to get the other. A regular octagon has exterior angle 45^°, so one interior angle is 180^° - 45^° = 135^°. You can check with the sum: (8 - 2) × 180^° = 1080^°, and 1080^° ÷ 8 = 135^°. The two routes agree.
The same link runs backwards. If a regular polygon has interior angle 156^°, the exterior angle is 180^° - 156^° = 24^°. The number of sides is 360^° ÷ 24^° = 15. Only then, if you need it, is the interior sum (15 - 2) × 180^° = 2340^°. Finding n from the exterior angle is quicker than trying to solve (n - 2) × 180^° ÷ n = 156^° from scratch.
Method: 1. Subtract an exterior angle from 180^° to get the interior angle. 2. A regular octagon has exterior angle 45^°, so one interior angle is 180-45=135^°. 3. From interior angle 156^°, the exterior angle is 24^° and the number of sides is 360÷ 24=15.
Read what is asked. “Sum of the interior angles” means (n - 2) × 180^°. “One interior angle” of a regular polygon means that sum divided by n, or 180^° minus the exterior angle. “One exterior angle” of a regular polygon means 360^° ÷ n. “How many sides” usually means divide 360^° by the exterior angle. If you are given the interior angle, subtract it from 180^° first so that you are dividing 360^° by the exterior angle.
A shape that is not regular still has an exterior sum of 360^° and an interior sum of (n - 2) × 180^°, but you cannot split either sum into equal pieces. A question that gives four angles of an irregular pentagon and asks for the fifth wants 540^° minus the four you have. A question that says “regular octagon” wants the equal split, exterior 45^° and interior 135^°.
Method: 1. Use (n-2)× 180^° when the question asks for the interior sum. 2. Use 360^°÷ n when it asks for one exterior angle of a regular polygon. 3. Subtract the interior angle from 180^° before you divide 360^° to find the number of sides.
Not every polygon in an exam is regular. When the sides and angles are different, you can still use the rule that the interior angles of an n-sided polygon add up to (n - 2) × 180^°. Add up the angles you are given, subtract from the total, and the difference is the missing angle. If some angles are given as expressions such as 3x or 2x + 20, add them all, set the total equal to the sum, and solve.
Write the sum in the first line of your working, for example "pentagon: (5 - 2) × 180 = 540". This line usually earns a mark by itself. Check the final answer by adding every angle again, and make sure the answer is sensible: a polygon with interior angles all under 180^° cannot have a missing angle bigger than that.
Method: 1. Count the sides n and work out (n - 2) × 180. 2. Add the angles you know, including any algebraic terms. 3. Subtract from the total, or solve the equation for x. 4. Add all the angles again to check they give the polygon total.
Tip: Common totals to remember: triangle 180^°, quadrilateral 360^°, pentagon 540^°, hexagon 720^°, octagon 1080^°.
For a regular polygon, every exterior angle is equal and all the exterior angles total 360^°. So the number of sides is n = 360 ÷ exterior angle. If you are given an interior angle, first turn it into an exterior angle with 180 - interior. For example, an interior angle of 150^° gives an exterior angle of 30^°, so n = 360 ÷ 30 = 12.
Another typical exam sentence is "each interior angle is 8 times the exterior angle". Call the exterior angle e. Then the interior angle is 8e and the two add up to 180^°, so 9e = 180 and e = 20. The polygon has 360 ÷ 20 = 18 sides. The number of sides must be a whole number, so a result such as 4.5 shows that such a regular polygon cannot exist.
Method: 1. Turn the information into the exterior angle e using interior + exterior = 180. 2. Calculate n = 360 ÷ e. 3. Check that n is a whole number. 4. If asked, find the sum of interior angles with (n - 2) × 180.
Tip: If the interior angle is a "ratio" of the exterior angle, call the exterior e and write the interior as a multiple of e.
A common exam picture shows two regular polygons that touch along a side, or several polygons around a point, and asks for an angle between them. Work out the interior angle of each polygon first. Then use the fact that angles at a point add up to 360^° to find the gap. For a regular hexagon the interior angle is 120^° and for a regular octagon it is 135^°.
Tessellation questions work the same way. Shapes fit round a point with no gaps only if their angles add up to exactly 360^°. Three regular hexagons give 3 × 120 = 360, so they fit. Regular octagons give 2 × 135 = 270, which leaves a gap, and 3 × 135 = 405, which is too big, so octagons alone cannot tessellate. Always show the sum when you explain.
Method: 1. Find the interior angle of every regular polygon in the picture. 2. Add the angles that meet at the point. 3. Subtract the total from 360^° to find the angle that is left. 4. For tessellations, check whether a whole number of angles makes exactly 360^°.
Drawing a diagonal in a regular polygon creates triangles, and the triangles are often isosceles because the polygon has equal sides. In a regular pentagon ABCDE, the triangle ABC has AB = BC and an apex angle of 108^°. The base angles are (180 - 108) ÷ 2 = 36^°. You can use this to find other angles at the same vertex.
Exams also ask "show that" and "is X correct?" questions. To show that no regular polygon has an interior angle of 100^°, find the exterior angle 80^° and calculate 360 ÷ 80 = 4.5, which is not a whole number. To show a claim is wrong, a single number check is enough. Always state the conclusion in words after the calculation.
Method: 1. Draw the diagonal and mark equal sides to spot isosceles triangles. 2. Use the interior angle as the apex of the triangle and find the base angles. 3. Use angles at a vertex to add or subtract the pieces. 4. For a "show that", calculate one key number and write the conclusion.
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