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Polygons

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Polygons

The big idea: The exterior angles of any polygon add up to 360°. From that one fact you get every interior angle of a regular polygon, and from the triangle sum you get the angle sum of any polygon.

Summary — What This Chapter Covers

Know the names: triangle, quadrilateral, pentagon, hexagon, octagon, and the special quadrilaterals.
The angles of a quadrilateral add up to 360°.
Know the side, angle and diagonal properties of the parallelogram, rectangle, square, rhombus, trapezium and kite.
A regular polygon has all sides equal and all angles equal: exterior angle = 360° ÷ n.
Angle sum of an n-sided polygon = (2n − 4) right angles = (n − 2) × 180°.
Congruent shapes are exactly the same shape and size.

1. Names and the Quadrilateral Angle Sum

Foundation & Higher

Polygons and quadrilaterals

Sides	3	4	5	6	8
Name	triangle	quadrilateral	pentagon	hexagon	octagon

A quadrilateral is any four-sided polygon. A diagonal splits it into two triangles, so its angles add up to 2 × 180° = 360°.

Worked Example

The four angles of a quadrilateral are 90°, (x + 15)°, (x + 25)° and (x + 35)°. Find the value of x.

2. Special Quadrilaterals

Foundation & Higher

Properties to know by heart

Shape	Sides	Angles	Diagonals
Parallelogram	opposite sides equal and parallel	opposite angles equal	bisect each other
Rectangle	opposite sides equal and parallel	all 90°	equal; bisect each other
Square	all equal; opposite sides parallel	all 90°	equal; bisect each other at 90°
Rhombus	all equal; opposite sides parallel	opposite angles equal	bisect each other at 90°
Trapezium	one pair of parallel sides	allied angles between the parallel sides add to 180°	—
Kite	two pairs of equal adjacent sides	one pair of opposite angles equal	cross at 90°; one bisects the other

Practice Question 1

ABCD is a parallelogram. Angle DAB = 68°. Find angle ABC and angle BCD.

3. Regular Polygons

Foundation & Higher

Interior and exterior angles

In a regular polygon every side is the same length and every angle is the same size. Walking round any polygon you turn through 360° in total, so the exterior angles add up to 360°.

Regular polygon with n sides exterior angle = 360° ÷ n interior angle = 180° − exterior angle n = 360° ÷ exterior angle

Worked Example

Work out the size of each interior angle of a regular octagon.

Practice Question 2

Each interior angle of a regular polygon is 156°. How many sides does it have?

4. Angle Sum of any Polygon

Foundation & Higher

(2n − 4) right angles

From one vertex, diagonals split an n-sided polygon into (n − 2) triangles. The spec writes the result as (2n − 4) right angles, which is the same thing.

Angle sum sum of interior angles = (n − 2) × 180° = (2n − 4) × 90°

Worked Example

A pentagon has angles 100°, 110°, 120°, x° and 2x°. Find x.

Quick table

Triangle 180° · quadrilateral 360° · pentagon 540° · hexagon 720° · octagon 1080°.

5. Congruence

Foundation & Higher

Memorise

Angle sum

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Polygons

Polygons

Polygons and quadrilaterals

Properties to know by heart

Interior and exterior angles

(2n − 4) right angles

Angle sum

Past papers that test Polygons

More Mathematics A topics