EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Interior angles of a polygon
Angles and trigonometry · Lesson 2 of 7
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what do the angles in a quadrilateral add up to?
360°
2. How many sides does a hexagon have?
6
3. Work out 8 × 180.
1440
4. Solve x/5 = 108.
x = 540
Learning Objectives
1. Name polygons up to 10 sides.
2. Work out the sum of the interior angles of any polygon.
3. Find a missing interior angle.
4. Work out the interior angle of a regular polygon.
Polygons
A polygon is a 2D shape with straight sides.
▸ Names. Triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.
▸ Interior angle. An angle inside the polygon, at a vertex (corner).
▸ Regular. All sides equal AND all angles equal.
▸ Irregular. Sides or angles not all equal - the angle sum is still the same.
Why the Formula Works
|
Draw lines from one vertex to every other vertex. A polygon with n sides always splits into n − 2 triangles, and each triangle holds 180°. So the interior angles add up to (n − 2) × 180°. |
An n-sided polygon splits into n − 2 triangles. |
Interior Angle Sums
|
Polygon |
Sides |
Triangles |
Angle sum |
|---|---|---|---|
|
Triangle |
3 |
1 |
180° |
|
Quadrilateral |
4 |
2 |
360° |
|
Pentagon |
5 |
3 |
540° |
|
Hexagon |
6 |
4 |
720° |
|
Octagon |
8 |
6 |
1080° |
|
Decagon |
10 |
8 |
1440° |
A Missing Angle in a Pentagon
|
Four of the angles of a pentagon are 100°, 120°, 95° and 110°. Work out the fifth angle. |
1. Angle sum of a pentagon
(5 − 2) × 180 = 540
2. Add the known angles
100 + 120 + 95 + 110 = 425
3. Subtract
540 − 425 = 115
Answer: 115°
Regular Polygons
In a regular polygon every interior angle is the same, so share the angle sum equally.
▸ Interior angle. Interior angle of a regular polygon = ((n − 2) × 180)/n.
▸ Regular hexagon. 720/6 = 120°.
▸ Regular octagon. 1080/8 = 135°.
▸ Growing. The more sides, the closer the interior angle gets to 180° - but it never reaches it.
The Interior Angle of a Regular Decagon
|
Work out the size of each interior angle of a regular decagon. |
1. A decagon has 10 sides
n = 10
2. Angle sum
(10 − 2) × 180 = 1440
3. Share equally between 10 angles
1440 ÷ 10 = 144
Answer: 144°
Working Back to the Number of Sides
|
Each interior angle of a regular polygon is 150°. How many sides does it have? |
1. Write the angle sum two ways
(n − 2) × 180 = 150n
2. Expand
180n − 360 = 150n
3. Solve
30n = 360, so n = 12
Answer: 12 sides (a dodecagon). Next lesson shows a quicker way.
Key Terms
|
Polygon A closed 2D shape with straight sides. |
Vertex A corner of a shape (plural: vertices). |
|
Interior angle An angle inside a polygon at a vertex. |
Regular polygon A polygon with all sides and all angles equal. |
|
Diagonal A straight line joining two vertices that are not next to each other. |
|
Your Task: Does It Tessellate?
10 minutes
|
A regular polygon tessellates (tiles a flat surface with no gaps) if copies of it fit exactly around a point. Work out the interior angles of the regular triangle, square, pentagon, hexagon and octagon, and decide which of them tessellate on their own. 1. Work out each interior angle. 2. Check whether it divides exactly into 360°. 3. Explain your answer. |
A good answer shows: Interior angles: 60, 90, 108, 120, 135. Only the triangle (6 × 60 = 360), square (4 × 90) and hexagon (3 × 120) divide exactly into 360°, so only those three tessellate.
Can I...?
☐ Name polygons up to 10 sides.
☐ Explain why the angle sum is (n − 2) × 180°.
☐ Find the angle sum of any polygon.
☐ Find a missing interior angle.
☐ Find the interior angle of a regular polygon.
☐ Find the number of sides from the interior angle.
Summary
✓ Angle sum = (n − 2) × 180°.
✓ Regular polygon: each interior angle = ((n − 2) × 180)/n.
✓ Missing angle: angle sum minus the angles you know.
|
EXAM FOCUS Work out the size of each interior angle of a regular nonagon (9 sides). (2 marks) Write the formula before you substitute. If you only remember one thing, remember that a polygon splits into n − 2 triangles - you can rebuild the formula from that. |