Polygons

A polygon is a shape made up of straight sides. Polygons include triangles, rectangles, parallelograms, pentagons, hexagons and so on. This sections explores the different types and properties of polygons:

• Regular polygons (shapes where each side is the same length and each angle is the same).
• Interior and exterior angles.
• Different types of triangle.
• Different types of quadrilateral.
• Other types of polygons.
• Star polygons.
• Symmetry.
• Tesselation (tiling shapes).

Regular Polygons

Find the hidden patterns in shapes with equal sides and equal angles.

A regular polygon has all its sides equal in length and all its interior angles equal. Equilateral triangles, squares, regular pentagons and regular hexagons are familiar examples.

You will investigate how the number of sides affects the size of the interior and exterior angles. You will learn to calculate these angles and discover why the exterior angles of a convex polygon add up to 360°.

These patterns lead to useful formulae that allow you to solve problems involving polygons with many different numbers of sides, without needing to measure each angle individually.

Triangles

Understand the properties of triangles and use them to solve problems.

Triangles are among the simplest geometric shapes, but their properties provide the foundation for many important mathematical ideas.

You will explore different types of triangles, including equilateral, isosceles, scalene, right-angled, acute and obtuse triangles. You will learn how their side lengths and angles determine their properties, and how to use the fact that their interior angles add up to 180°.

You will also investigate relationships between equal sides and equal angles, as well as other important triangle properties.

A strong understanding of triangles will help you with angle calculations, geometric proofs, congruence, similarity and trigonometry.

Quadrilaterals

Learn to recognise four-sided shapes and understand what makes each one special.

A quadrilateral is a polygon with four sides. Although quadrilaterals can take many forms, familiar examples such as squares, rectangles, parallelograms, rhombuses, trapeziums and kites all have distinctive properties.

You will explore how these shapes differ in their side lengths, angles, parallel sides and lines of symmetry. You will also learn that the interior angles of any quadrilateral add up to 360°.

Recognising the relationships between quadrilaterals helps you identify shapes from their properties, calculate missing angles and solve problems that involve more than one geometric rule.

More Quadrilaterals

Look beyond the basic shapes to discover more specialised quadrilaterals.

Some quadrilaterals have additional properties that make them particularly interesting. For example, a kite has two pairs of adjacent equal sides, while different types of trapezium have their own combinations of parallel sides, angles and symmetry.

You will explore how quadrilaterals can be classified according to their properties and how one type of shape can be a special case of another. For example, a square is also a rectangle, a rhombus and a parallelogram.

Understanding these connections helps you move beyond simply memorising shape names. You will develop a clearer picture of how geometric properties fit together and how to use them in mathematical reasoning.

Interior and Exterior Angles

Calculate angles in polygons using patterns and formulae.

Interior angles are the angles inside a polygon, while exterior angles are formed when one of its sides is extended. The relationship between these angles creates useful patterns that work for polygons of different sizes.

You will learn how to calculate the sum of the interior angles of a polygon using the formula:

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

Here, n represents the number of sides.

You will also discover that the exterior angles of a convex polygon add up to 360°. For regular polygons, you can use this fact to calculate each exterior angle and then work out each interior angle.

These formulae allow you to solve problems involving pentagons, hexagons, octagons and polygons with any number of sides.

Related articles

• Regular polygons
• Interior and exterior angles of a polygon
• Triangles
• Quadrilaterals
• Quadrilateral family tree
• Pentagons - polygons with 5 sides
• Hexagons - 6-sided polygon: angles, symmetry and properties
• Heptagons - polygons with 7 sides
• Octagons - polygons with 8 sides
• Nonagons - polygons with 9 sides
• Decagons - polygons with 10 sides
• Hendecagons - polygons with 11 sides
• Dodecagons - polygons with 12 sides
• n-gons - polygons with any number of sides
• Other types of polygon
• Star polygons