3 sides
Triangle
Learning Objectives
- name polygons with up to ten sides;
- calculate the sum of interior angles;
- find one interior or exterior angle of a regular polygon;
- use exterior angles to find the number of sides;
- solve algebraic polygon problems.
Introduction
This lesson develops Polygons from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
What Is a Polygon?
A polygon is a closed plane shape made only from straight line segments.
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
7 sides
Heptagon
8 sides
Octagon
9 sides
Nonagon
10 sides
Decagon
Interior Angle Sum
Interior angle sum = (n − 2) × 180°
Regular Polygons
A regular polygon has equal sides and equal interior angles.
One exterior angle = 360° ÷ n
One interior angle = 180° − exterior angle
Worked Examples
Find the interior-angle sum of an octagon.
(8−2)×180=1080°.
Find one exterior angle of a regular nonagon.
360÷9=40°.
A regular polygon has exterior angle 24°. Find its number of sides.
n=360÷24=15.
Try It: Explore Regular Polygons
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Name polygons with 5, 6, 8 and 10 sides.
- Find the interior-angle sum of a hexagon.
- Find one exterior angle of a regular octagon.
- Find one interior angle of a regular pentagon.
- A regular polygon has exterior angle 30°. Find its number of sides.
- The interior-angle sum is 1440°. Find the number of sides.
- The interior angles of a pentagon are x, x+10, x+20, x+30 and x+40. Find all five angles.
✅ Check Your Work
Show answers and working
- Pentagon, hexagon, octagon and decagon.
- (6−2)×180=720°.
- 360÷8=45°.
- Exterior angle=72°, so interior angle=108°.
- 12 sides.
- (n−2)×180=1440, so n=10.
- 5x+100=540, so x=88°. Angles: 88°, 98°, 108°, 118°, 128°.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Interior sum=(n−2)×180°.
- Exterior angles total 360°.
- Regular polygons have equal sides and equal angles.