0580

Angles in Polygons and Parallel Lines

Geometry

Interior and exterior angles

  • An interior angle sits inside the polygon at a corner
  • An exterior angle sits between one side and the extension of the next side
  • At every corner the interior and exterior angles lie on a straight line, so they add to 180°
  • Whatever the polygon and however many sides it has, its exterior angles total 360°
  • For the inside angles the total depends on n: a polygon with n sides splits into n − 2 triangles, so they come to 180° × (n − 2)
    • A triangle gives 180°, a quadrilateral 360°, a pentagon 540° and a hexagon 720°
An irregular hexagon with each of its six interior angles labelled a to f in blue, and each side extended with a dashed line so the exterior angle beside it, p to u, is marked in red, the interior and exterior angle at each corner lying on a straight line
Source: Angles in polygons by Save My Exams

Regular polygons

  • A regular polygon has all sides equal and all angles equal, so its angles can be found by dividing
  • Each exterior angle is 360° ÷ n, and this is usually the quicker route
  • Each interior angle is then 180° minus that exterior angle
  • The same result comes from 180(n − 2) ÷ n, so either method is acceptable
  • Working backwards, n = 360° ÷ exterior angle, which is how a question asks for the number of sides
Regular polygonSidesInterior sumEach interiorEach exterior
Equilateral triangle3180°60°120°
Square4360°90°90°
Pentagon5540°108°72°
Hexagon6720°120°60°
Octagon81080°135°45°
Decagon101440°144°36°
Exam tip

Write the reason out in full — "alternate angles are equal", "co-interior angles add to 180°". Mark schemes accept the name with or without the "are equal" ending, but reject abbreviations such as "alt" or "correspond" outright, and reject "Z-angles" and "F-angles" unless the proper name is there too. The exterior angles of every polygon add to 360°, however many sides it has.

Worked example

Finding the number of sides

Each exterior angle of a regular polygon is 24°. Work out the number of sides, and the size of each interior angle.

Solution:

  • The exterior angles of any polygon add to 360°
  • For a regular polygon they are all equal, so the number of sides is 360 ÷ 24
  • The polygon has 15 sides
  • The interior and exterior angles at each corner lie on a straight line
  • Each interior angle is 180 − 24 = 156°
  • Check with the other method: 180 × (15 − 2) = 2340, and 2340 ÷ 15 = 156

Irregular polygons

  • The interior angle sum formula works for any polygon, so an irregular one with a missing angle is found by subtraction
  • Add every angle you are given, then take the total from 180° × (n − 2)
    • Five angles of a hexagon are 100°, 145°, 98°, 130° and 122°, which total 595°
    • A hexagon's interior angles total 720°, so the missing angle is 720 − 595 = 125°
  • The exterior angles still add to 360° whether the polygon is regular or not