Understanding Quadrilaterals
Focused last-minute revision with key concepts, formulas, worked examples, exam traps and practice.
Chapter Snapshot
- Interior angle sum of an n-sided polygon is (n-2)×180°.
- Exterior angles of any convex polygon add to 360°.
- Angles of a quadrilateral add to 360°.
- Opposite sides and angles of a parallelogram are equal.
- A square is both a rectangle and a rhombus.
What You Will Learn
- Identify polygons and diagonals
- Use the polygon interior-angle sum formula
- Use the exterior-angle sum property
- Classify quadrilaterals
- Recall key properties of special quadrilaterals
Last-Minute Chapter Map
Use this chapter in three passes:
- Understand the core ideas.
- Memorise formulas and exam traps.
- Attempt the compact practice set without looking at the answers.
Polygons
A polygon is a closed figure made only of line segments. A diagonal joins two non-adjacent vertices. Triangles, quadrilaterals, pentagons and hexagons are common examples.
Angle sums
The interior angle sum depends only on the number of sides. For a quadrilateral the sum is 360°. The sum of one exterior angle at each vertex of any convex polygon is always 360°.
Parallelogram family
A parallelogram has both pairs of opposite sides parallel and equal, opposite angles equal and diagonals that bisect each other. Rectangles, rhombi and squares are special parallelograms.
Special quadrilaterals
A rectangle has four right angles and equal diagonals. A rhombus has four equal sides and perpendicular bisecting diagonals. A square has four equal sides, four right angles and equal perpendicular bisecting diagonals. A trapezium has at least one pair of parallel sides; a kite has two pairs of adjacent equal sides.
Special quadrilaterals are classified by side, angle and diagonal properties.
Formula Sheet
Formula / Key Relations
Interior angle sum
n is the number of sides.
Exterior angle sum
For one exterior angle at each vertex.
Each exterior angle of a regular polygon
All exterior angles are equal in a regular polygon.
Worked Examples
Use n = 5.
Exterior angles are equal.
Exam Traps
Quick Revision
Quick Revision
- Interior angle sum of an n-sided polygon is (n-2)×180°.
- Exterior angles of any convex polygon add to 360°.
- Angles of a quadrilateral add to 360°.
- Opposite sides and angles of a parallelogram are equal.
- A square is both a rectangle and a rhombus.
- Revise the formula sheet once more before attempting questions.
Compact Practice
Interior angle sum of a pentagon?
View Solution
540°
Four equal sides but angles not necessarily 90°?
View Solution
Rhombus
Each exterior angle of a regular 12-gon?
View Solution
30°
Three angles of a quadrilateral are 90°, 110°, 80°. Find the fourth.
View Solution
80°
Final Chapter Summary
Chapter Summary
- Interior angle sum of an n-sided polygon is (n-2)×180°.
- Exterior angles of any convex polygon add to 360°.
- Angles of a quadrilateral add to 360°.
- Opposite sides and angles of a parallelogram are equal.
- A square is both a rectangle and a rhombus.
- Prioritise correct method, clean working and final units where applicable.