Interior and Exterior Angles of Polygons — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers the angles inside and outside a polygon. By the end of this guide you should be able to find the sum of the interior angles of any polygon, find each angle of a regular polygon, and use the fact that exterior angles always total 360°.
You should also be able to work backwards from an angle to the number of sides, find a missing angle in an irregular polygon, and explain which regular polygons tessellate.
The organising idea is that the exterior angles of every polygon add to 360°, no matter how many sides it has. Walk all the way round the outside of any polygon and you turn through one full revolution by the time you are back where you started — a triangle turns through 360° in three big turns, a twenty-sided shape in twenty small ones, but always 360° in total. That single fact is usually the quickest route into a problem, and because interior and exterior angles sit on a straight line together, it gives you the interior angles too.
Key terms and definitions
Polygon — a closed shape with straight sides.
Regular polygon — one with all sides equal and all angles equal.
Irregular polygon — one whose sides or angles are not all equal.
Interior angle — the angle inside the polygon at a vertex.
Exterior angle — the angle between one side and the extension of the next side.
Vertex — a corner of the polygon. The plural is vertices.
Tessellate — to fit together with no gaps and no overlaps.
Core concepts
Naming polygons
Learning the names saves time: triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9) and decagon (10).
Beyond those, a polygon is usually described by its number of sides, such as a 15-sided polygon.
The sum of the interior angles
Any polygon can be split into triangles by drawing diagonals from one vertex, and each triangle contributes 180°.
A polygon with n sides splits into n − 2 triangles, so:
sum of interior angles = (n − 2) × 180°
A quadrilateral gives (4 − 2) × 180° = 360°, which matches the familiar result. A pentagon gives 540°, a hexagon 720°, and an octagon 1080°.
This formula works for every polygon, regular or not. The angles of an irregular pentagon still total 540°, however uneven they are.
Exterior angles always total 360°
For any polygon, the exterior angles add to 360°.
The number of sides makes no difference, which is what makes this fact so useful. Many problems that look complicated collapse to a single division once you use it.
For a regular polygon, all the exterior angles are equal, so:
each exterior angle = 360° ÷ n
A regular hexagon therefore has exterior angles of 360 ÷ 6 = 60°, and a regular decagon 36°.
Interior and exterior together
At each vertex the interior and exterior angles lie on a straight line, so:
interior + exterior = 180°
Knowing one gives the other immediately. A regular hexagon with an exterior angle of 60° has interior angles of 180 − 60 = 120°.
This pairing is what makes the exterior-angle route so efficient. Rather than computing (n − 2) × 180° and dividing by n, you can divide 360° by n and subtract from 180°, which is usually easier arithmetic and reaches the same answer.
Each interior angle of a regular polygon
Two routes, both correct.
Divide the total by the number of angles: (n − 2) × 180° ÷ n. For a regular hexagon, 720 ÷ 6 = 120°.
Or go via the exterior angle: 360 ÷ 6 = 60°, then 180 − 60 = 120°.
The second is faster and less error-prone, particularly for polygons with many sides.
Finding the number of sides
Questions frequently run this backwards, giving an angle and asking how many sides the polygon has. Work through the exterior angle.
If each interior angle is 150°, then each exterior angle is 180 − 150 = 30°, and the number of sides is 360 ÷ 30 = 12.
Going via the interior formula instead means solving (n − 2) × 180 ÷ n = 150, which is an equation with n on both top and bottom — perfectly possible, but much slower.
A useful check: 360 must divide exactly by the exterior angle. If it does not, no regular polygon has that angle, and some questions test exactly this.
Missing angles in irregular polygons
For an irregular polygon, use the interior sum. Add the angles you know and subtract from the total.
In a pentagon with four angles of 100°, 120°, 90° and 130°, those total 440°, so the fifth angle is 540 − 440 = 100°.
The exterior angles of an irregular polygon still total 360°, so a missing exterior angle can be found the same way.
Problems combining several polygons
Harder questions fit two or more regular polygons together around a point, or against one another along an edge, and ask for the angle in the gap.
The method is always the same: work out the interior angle of each polygon separately, then use the fact that angles round a point add to 360°.
Suppose a regular hexagon and a regular octagon meet at a point, with a gap between them. The hexagon contributes 120° and the octagon contributes 180 − (360 ÷ 8) = 135°. Together they take up 255°, so the gap is 360 − 255 = 105°.
Where polygons meet along a straight edge instead, the relevant total is 180° rather than 360°.
Label each angle on the diagram as you calculate it. These questions carry several marks and are marked on the reasoning, so an unlabelled final number scores poorly even when it is right.
Angles in a polygon split into triangles
The formula came from splitting the polygon into triangles, and questions sometimes ask you to explain that rather than just use it.
Drawing every diagonal from one vertex of an n-sided polygon produces n − 2 triangles: the two sides meeting at that vertex cannot form triangles of their own, which is where the subtraction of 2 comes from.
Since each triangle's angles total 180° and every angle of every triangle lies inside the polygon, the polygon's angles total (n − 2) × 180°.
Being able to state that reasoning is worth a mark in "explain why" questions, which appear regularly on this topic.
Tessellation
A shape tessellates if copies of it fit round a point with no gaps, which requires its interior angle to divide exactly into 360°.
Only three regular polygons manage it: the equilateral triangle (60° — six fit round a point), the square (90° — four fit), and the regular hexagon (120° — three fit).
A regular pentagon has interior angles of 108°, and 360 ÷ 108 is not a whole number, so pentagons leave gaps. Anything with more sides has an interior angle above 120°, so fewer than three fit and the shape cannot tessellate either.
Worked examples
Example 1: Angles of a regular polygon
Find the size of each interior angle of a regular 15-sided polygon.
Start with the exterior angle, which is the quicker route: 360 ÷ 15 = 24°.
The interior angle is on a straight line with it: 180 − 24 = 156°.
Checking by the other method: (15 − 2) × 180 = 2340°, and 2340 ÷ 15 = 156°. ✓ The two routes agree, but the first needed only two short calculations.
Example 2: Working backwards to the number of sides
Each interior angle of a regular polygon is 162°. How many sides does it have?
The exterior angle is 180 − 162 = 18°.
The number of sides is 360 ÷ 18 = 20.
The division comes out exactly, which confirms that such a polygon exists. Had the question given 145°, the exterior angle would be 35°, and 360 ÷ 35 is not a whole number — so no regular polygon has that interior angle.
Example 3: A missing angle in an irregular polygon
An irregular hexagon has five angles measuring 110°, 130°, 95°, 140° and 125°. Find the sixth.
The interior angles of a hexagon total (6 − 2) × 180 = 720°.
The five given angles add to 110 + 130 + 95 + 140 + 125 = 600°.
The sixth angle is 720 − 600 = 120°.
Being irregular changes nothing about the total; it only means the angles are not all equal.
Common mistakes and how to avoid them
Using (n − 2) × 180° for one angle. That formula gives the sum. Divide by n for a single angle, and only in a regular polygon.
Dividing the interior sum by n for an irregular polygon. The angles are not equal, so there is no single value to find.
Thinking the exterior sum depends on the number of sides. It is always 360°.
Forgetting that interior and exterior add to 180°. This is the link between the two halves of the topic.
Taking the long route to find the number of sides. Go via the exterior angle and one division.
Assuming every angle gives a valid polygon. 360 must divide exactly by the exterior angle.
Assuming all regular polygons tessellate. Only the triangle, square and hexagon do.
Exam technique for "Interior and Exterior Angles"
Reach for the exterior angle first. Most questions in this topic are one division away once you do.
Write the formula you are using before substituting, since method marks are attached to it.
State clearly whether you have found a sum or a single angle — muddling the two is the commonest reason for a correct calculation earning no marks.
Check that a division by the exterior angle comes out exactly. A fractional number of sides means an error, or a deliberately impossible angle.
For problems combining polygons, work out the interior angle of each and use angles round a point totalling 360°.
Give answers in degrees with the symbol, and show each stage on its own line.
Quick revision summary
Exterior angles always total 360°, whatever the number of sides. This is usually the fastest way into a problem.
Sum of interior angles = (n − 2) × 180°, because the polygon splits into n − 2 triangles. It holds for regular and irregular polygons alike.
At each vertex, interior + exterior = 180°.
For a regular polygon: each exterior angle = 360° ÷ n, and each interior angle = 180° − that. A regular hexagon gives 60° and 120°.
To find the number of sides from an interior angle, subtract from 180° to get the exterior angle, then divide 360° by it. The division must come out exactly.
For an irregular polygon, add the known angles and subtract from the total.
Only the equilateral triangle, square and regular hexagon tessellate, because only 60°, 90° and 120° divide exactly into 360°.