6.5 More Exterior and Interior Angles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: finding the number of sides, impossible polygons, shapes that meet at a point and combined shapes. 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

• Find the number of sides of a regular polygon from its exterior or interior angle
• Find the number of sides from the sum of the interior angles
• Decide whether a polygon with a given angle or angle sum can exist
• Solve problems with polygons that meet at a point or share a side

Key words

• Exterior angle: Each exterior angle of a regular n-sided polygon is 360° ÷ n.
• Interior angle: Interior angle = 180° − exterior angle.
• Tessellate: Shapes tessellate if they fit together with no gaps or overlaps.
• Vertex: A corner of a shape (plural: vertices).

Finding the number of sides

Turn the rule round: if each exterior angle of a regular polygon is e, then the number of sides is n = 360 ÷ e.

Worked example: from the exterior angle

Each exterior angle of a regular polygon is 24°. How many sides does it have?
n = 360 ÷ 24 = 15

Answer: 15 sides

Worked example: from the interior angle

Each interior angle of a regular polygon is 140°. How many sides does it have?
Exterior angle = 180 − 140 = 40°. n = 360 ÷ 40 = 9

Answer: 9 sides (a nonagon)

Worked example: from the angle sum

The interior angles of a polygon add up to 1260°. How many sides does it have?
(n − 2) × 180 = 1260, so n − 2 = 7 and n = 9

Answer: 9 sides

Is it possible?

The number of sides must be a whole number. So for a regular polygon the exterior angle must divide exactly into 360, and for any polygon the angle sum must be a multiple of 180.

Worked example: can a regular polygon have an interior angle of 130°?

Exterior angle = 180 − 130 = 50°. 360 ÷ 50 = 7.2
7.2 is not a whole number.

Answer: no

Key fact: The exterior angle of a regular polygon can be at most 120° (triangle), so the interior angle is at least 60° and always less than 180°.

Shapes meeting at a point

Angles around a point add up to 360°. Regular polygons tessellate on their own only if their interior angle divides exactly into 360.

• Equilateral triangle; Interior angle 60°; Tessellates on its own? yes: 6 × 60° = 360°
• Square; Interior angle 90°; Tessellates on its own? yes: 4 × 90° = 360°
• Regular pentagon; Interior angle 108°; Tessellates on its own? no: 360 ÷ 108 is not whole
• Regular hexagon; Interior angle 120°; Tessellates on its own? yes: 3 × 120° = 360°
• Regular octagon; Interior angle 135°; Tessellates on its own? no, but with squares: 135° + 135° + 90° = 360°

Polygons and triangles together

Diagonals of a regular polygon often make isosceles triangles, because the sides of the polygon are equal.

Worked example: an angle made by a diagonal

Find angle BAC.
Angle ABC = 120° (interior angle of a regular hexagon).
Triangle ABC is isosceles, so angle BAC = (180 − 120) ÷ 2 = 30°

Answer: 30°

Common mistakes

• Dividing 360 by the interior angle to find the number of sides. Use the exterior angle.
• Giving a non-whole number of sides, e.g. 7.2. That means the polygon cannot exist.
• Forgetting that the base angles of the isosceles triangle are equal.
• Using 180° instead of 360° for angles that meet at a point.

Revision questions

Try these, then check your answers below each one.

Question 1. Each exterior angle of a regular polygon is 40°. How many sides does it have?

Answer: 9

Working: 360 ÷ 40 = 9

Question 2. Each interior angle of a regular polygon is 165°. How many sides does it have?

Answer: 24

Working: Exterior = 180 - 165 = 15°. 360 ÷ 15 = 24 sides.

Question 3. The interior angles of a polygon add up to 1980°. How many sides does it have?

Answer: 13

Working: 1980 ÷ 180 = 11, so n - 2 = 11 and n = 13.

Related lessons

• Previous lesson: 6.4 Exterior and Interior Angles
• Next lesson: 6.6 Geometrical Problems
• In this chapter: 6.1 Properties of Shapes
• In this chapter: 6.2 Angles in Parallel Lines
• In this chapter: 6.3 Angles in Triangles

Continue in the full course

This lesson is section 6.5 (EGCMF0605) 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.