6.4 Exterior and Interior Angles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: naming polygons, the sum of the interior angles, and interior and exterior angles of regular polygons. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Name polygons and say whether they are regular or irregular
• Work out the sum of the interior angles of a polygon
• Work out the interior and exterior angles of a regular polygon
• Use the facts that exterior angles add up to 360° and interior + exterior = 180°

Key words

• Polygon: A flat shape with straight sides. An n-sided polygon has n angles.
• Regular polygon: All sides are equal and all angles are equal.
• Irregular polygon: Sides or angles are not all equal.
• Interior angle: An angle inside a polygon, where two sides meet.
• Exterior angle: The angle between one side and the next side made longer. Interior + exterior = 180°.

Naming polygons

• 3; Name triangle; Sides 7; Name heptagon
• 4; Name quadrilateral; Sides 8; Name octagon
• 5; Name pentagon; Sides 9; Name nonagon
• 6; Name hexagon; Sides 10; Name decagon

A regular polygon has equal sides and equal angles. A square is a regular quadrilateral; a rectangle is not, because its sides are not all equal.

The sum of the interior angles

From one corner, draw diagonals to all the other corners. A polygon with n sides splits into n − 2 triangles.

Key fact: Sum of interior angles = (n − 2) × 180°

• Angle sum; 3 180°; 4 360°; 5 540°; 6 720°; 8 1080°; 10 1440°; 12 1800°

Worked example: a missing angle

A hexagon has angles 115°, 130°, 98°, 140°, 122° and x.
Sum = (6 − 2) × 180 = 720°. 115 + 130 + 98 + 140 + 122 = 605. x = 720 − 605 = 115

Answer: x = 115°

Interior angles of regular polygons

All the angles of a regular polygon are equal, so divide the angle sum by the number of angles.

Worked example: a regular octagon

Sum = (8 − 2) × 180 = 1080°. Each angle = 1080 ÷ 8 = 135°

Answer: 135°

Exterior angles

Key fact: The exterior angles of any polygon add up to 360°. For a regular polygon, each exterior angle = 360° ÷ n. Interior angle + exterior angle = 180°.

Worked example: a regular decagon

Exterior angle = 360 ÷ 10 = 36°
Interior angle = 180 − 36 = 144°

Answer: exterior 36°, interior 144°

Worked example: an irregular polygon

Four of the five exterior angles of a pentagon are 62°, 85°, 70° and 58°. Find the fifth.
62 + 85 + 70 + 58 = 275. 360 − 275 = 85

Answer: 85°

• Equilateral triangle; Exterior angle 120°; Interior angle 60°
• Square; Exterior angle 90°; Interior angle 90°
• Pentagon; Exterior angle 72°; Interior angle 108°
• Hexagon; Exterior angle 60°; Interior angle 120°
• Octagon; Exterior angle 45°; Interior angle 135°
• Decagon; Exterior angle 36°; Interior angle 144°

Common mistakes

• Using n × 180 instead of (n − 2) × 180 for the angle sum.
• Dividing the angle sum by n for an irregular polygon. Only regular polygons have equal angles.
• Thinking the exterior angles add up to more for a bigger polygon. They always add up to 360°.
• Mixing up interior and exterior: the exterior angle is outside, between a side and the next side extended.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the sum of the interior angles of a heptagon (7 sides).

Answer: 900 °

Working: (7 - 2) × 180 = 900°

Question 2. Work out the size of each exterior angle of a regular nonagon (9 sides).

Answer: 40 °

Working: 360 ÷ 9 = 40°

Question 3. Work out the size of each interior angle of a regular 12-sided polygon.

Answer: 150 °

Working: Exterior = 360 ÷ 12 = 30°. Interior = 180 - 30 = 150°.

Related lessons

• Previous lesson: 6.3 Angles in Triangles
• Next lesson: 6.5 More Exterior and Interior Angles
• In this chapter: 6.1 Properties of Shapes
• In this chapter: 6.2 Angles in Parallel Lines
• In this chapter: 6.6 Geometrical Problems

Continue in the full course

This lesson is section 6.4 (EGCMF0604) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.