Quadrilaterals (AQA GCSE Maths): Revision Notes
Quadrilaterals
What is a quadrilateral?
A quadrilateral is any shape that has exactly four sides. These shapes are fundamental in geometry and appear everywhere around us, from windows and doors to playing cards and book covers.
Every quadrilateral, regardless of its type, must have exactly four sides and four vertices (corners). This is the defining characteristic that makes it a quadrilateral.
The diagonals of a quadrilateral are the straight lines that connect opposite corners (vertices) of the shape. Every quadrilateral has two diagonals, and these help us understand the properties of different types of quadrilaterals.
Types of quadrilaterals
Understanding the different types of quadrilaterals means learning about their unique properties. Each type has specific characteristics regarding their sides, angles, diagonals, and symmetry.
Square
A square is the most regular quadrilateral, where everything is perfectly equal and orderly.
Properties:
- All four sides are exactly the same length
- All four angles are right angles ()
- Diagonals are equal in length and bisect each other at right angles
- Has 4 lines of symmetry (vertical, horizontal, and both diagonal lines)
- Has rotational symmetry of order 4 (looks identical when rotated by , , , and )
The word bisect means to cut exactly in half, which is what the diagonals do to each other in a square.
Rectangle
A rectangle is like a stretched or compressed square, maintaining right angles but not equal sides.
Properties:
- Opposite sides are parallel and equal in length
- All four angles are right angles ()
- Diagonals are equal in length and bisect each other (but not at right angles)
- Has 2 lines of symmetry (vertical and horizontal through the centre)
- Has rotational symmetry of order 2 (looks identical when rotated )
Parallelogram
A parallelogram gets its name from having parallel sides, but unlike rectangles, the angles aren't necessarily .
Properties:
- Opposite sides are parallel and equal in length
- Opposite angles are equal (but not necessarily )
- Diagonals bisect each other (but are not equal in length)
- Has no lines of symmetry
- Has rotational symmetry of order 2
Remember that parallelograms don't have right angles unless they are special cases like rectangles or squares. This is a common mistake in exams!
Trapezium
A trapezium is the simplest type of quadrilateral in terms of parallel sides.
Properties:
- Has exactly one pair of parallel sides (called bases)
- The other pair of sides are not parallel
- Generally has no special angle, diagonal, or symmetry properties
Kite
A kite gets its name from its resemblance to the flying toy, with a distinctive symmetrical appearance.
Properties:
- Has two pairs of adjacent sides that are equal (sides next to each other, not opposite)
- Has one pair of opposite angles that are equal
- Diagonals cross each other at right angles ()
- Has 1 line of symmetry (running through the vertex where unequal sides meet)
In a kite, "adjacent sides" means sides that are next to each other, sharing a vertex. This is different from "opposite sides" which are across from each other.
Rhombus
A rhombus can be thought of as a "squashed square" - all sides are equal but the angles aren't necessarily .
Properties:
- All four sides are equal in length
- Opposite sides are parallel
- Opposite angles are equal
- Diagonals bisect each other at right angles ()
- Has 2 lines of symmetry (both diagonal lines)
- Has rotational symmetry of order 2
Exam tips for identifying quadrilaterals
Critical Exam Strategy:
When answering exam questions about quadrilaterals, follow these essential steps:
- Read the properties carefully - look for key words like parallel, equal, right angle, and bisect
- Check all properties - a shape must satisfy ALL the stated properties to be correctly identified
- Remember the hierarchy - squares are special rectangles, and rectangles are special parallelograms
- Use process of elimination - rule out shapes that don't fit the given properties
Common mistakes include confusing adjacent vs opposite sides, and forgetting that some quadrilaterals are special cases of others.
Key Points to Remember:
- Quadrilaterals are four-sided shapes, and diagonals connect opposite corners
- Squares have everything equal: sides, angles, and perpendicular diagonals
- Rectangles have right angles and equal opposite sides, but not all sides equal
- Parallelograms have parallel opposite sides and equal opposite angles
- Kites have two pairs of equal adjacent sides and perpendicular diagonals
- Rhombuses have all equal sides and perpendicular diagonals, like "squashed squares"