EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Exterior angles of a polygon
Angles and trigonometry · Lesson 3 of 7
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what do the interior angles of a pentagon add up to?
540°
2. What do angles on a straight line add up to?
180°
3. Work out 360 ÷ 8.
45
4. What is each interior angle of a regular hexagon?
120°
Learning Objectives
1. Know that the exterior angles of any polygon add up to 360°.
2. Find the exterior angle of a regular polygon.
3. Use interior angle + exterior angle = 180°.
4. Find the number of sides of a regular polygon from one of its angles.
A Full Turn
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An exterior angle is the angle between one side and the extension of the next side. Walk round the shape and you turn by each exterior angle in turn - and arrive facing the way you started. That is one full turn, so the exterior angles add up to 360° for every polygon. |
The exterior angles fit together to make one full turn. |
Exterior Angle Facts
Two facts do almost all the work.
▸ The sum. The exterior angles of any polygon add up to 360°.
▸ Interior and exterior. At each vertex, interior angle + exterior angle = 180° (angles on a straight line).
▸ Regular polygon. Each exterior angle = 360/n.
▸ Number of sides. n = 360/(exterior angle).
Angles of a Regular Pentagon
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Work out the exterior and interior angles of a regular pentagon. |
1. Exterior angle
360 ÷ 5 = 72°
2. Interior angle: they add up to 180°
180 − 72 = 108°
3. Check with last lesson's formula
540 ÷ 5 = 108°
Answer: Exterior 72°, interior 108°
How Many Sides?
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Each interior angle of a regular polygon is 160°. How many sides does it have? |
1. Find the exterior angle
180 − 160 = 20°
2. Exterior angles add up to 360°
360 ÷ 20 = 18
Answer: 18 sides
Which Method?
Exterior angles are usually quicker for regular polygons.
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1 Regular, angles wanted Exterior = 360 ÷ n, then interior = 180 − exterior. |
2 Regular, sides wanted Exterior = 180 − interior, then n = 360 ÷ exterior. |
3 Irregular Use the angle sum (n − 2) × 180, or exterior angles adding up to 360°. |
4 Shapes joined together Use angles around a point: they add up to 360°. |
Polygons Meeting at a Point
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A regular hexagon and a square share a side, and meet at a point. Work out the angle x in the gap between them at that point. |
1. Interior angle of a regular hexagon
180 − 360 ÷ 6 = 120°
2. Interior angle of a square
90°
3. Angles around a point add up to 360°
x = 360 − 120 − 90 = 150
Answer: x = 150°
Key Terms
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Exterior angle The angle between one side of a polygon and the extension of the side next to it. |
Interior angle The angle inside a polygon at a vertex. |
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Angles around a point Angles meeting at a point; they add up to 360°. |
Regular polygon A polygon with all sides and all angles equal. |
Your Task: Impossible Polygons
10 minutes
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For each angle, decide whether it could be the exterior angle of a regular polygon. If it could, say how many sides the polygon has: 40°, 50°, 24°, 70°, 15°, 1°. 1. Divide 360 by the angle. 2. A whole number means it is possible. 3. Write down the number of sides. |
A good answer shows: 40°: 9 sides. 50°: no (7.2). 24°: 15 sides. 70°: no. 15°: 24 sides. 1°: 360 sides. An exterior angle works only if it divides exactly into 360.
Can I...?
☐ Recall that exterior angles add up to 360°.
☐ Find the exterior angle of a regular polygon.
☐ Use interior + exterior = 180°.
☐ Find the number of sides from an angle.
☐ Decide whether an angle is possible for a regular polygon.
☐ Solve problems with polygons meeting at a point.
Summary
✓ Exterior angles of any polygon add up to 360°.
✓ Regular polygon: exterior angle = 360 ÷ n, and n = 360 ÷ exterior angle.
✓ Interior + exterior = 180°.
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EXAM FOCUS The diagram shows a regular hexagon and a square that share a side. Work out the size of angle x. (3 marks) For any regular polygon question, work out the exterior angle first: 360 ÷ n. Everything else follows from it. |