Edexcel International GCSE Mathematics A 4MA1: Geometry and trigonometry – Revision Notes

For full explanations and worked examples, use the Geometry and trigonometry study guide. These notes condense sections 4.1–4.8 of Topic 4 in the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series examined on it. Sections 4.1–4.5 are the same on both tiers; parts of 4.6–4.8 are Higher tier only and are labelled. A calculator may be used on every paper.

Course hub: Edexcel IGCSE Mathematics. Tick off each statement on the printable checklist, then try the practice questions or a free diagnostic.

Angle facts to quote (4.1 and 4.7)

Use these exact wordings as reasons.

Fact	Reason to write
Angles on a straight line	add up to 180°
Angles at a point	add up to 360°
Vertically opposite angles	are equal
Alternate angles (parallel lines)	are equal
Corresponding angles (parallel lines)	are equal
Allied / co-interior angles (parallel lines)	add up to 180°
Angles in a triangle	add up to 180°
Exterior angle of a triangle	equals the sum of the interior opposite angles
Isosceles triangle	base angles are equal
Angles in a quadrilateral	add up to 360°

Angle types: acute < 90°, right = 90°, obtuse between 90° and 180°, reflex between 180° and 360°.

Polygons (4.2) and symmetry (4.3)

Sum of interior angles     = (2n − 4) × 90°   (the same as (n − 2) × 180°)
Exterior angle (regular)   = 360° ÷ n
Interior angle (regular)   = 180° − exterior angle
Number of sides (regular)  = 360° ÷ exterior angle

Shape	Lines of symmetry	Rotational order
Square	4	4
Rectangle	2	2
Rhombus	2	2
Parallelogram	0	2
Kite	1	1
Isosceles trapezium	1	1
Regular n-gon	n	n

Congruent means the same shape and the same size (reflections included).

Measures (4.4)

• Bearings: from north, clockwise, three figures. Back bearing = bearing ± 180°.
• Time: count up to the next whole hour, then whole hours, then the remaining minutes. Convert minutes to hours by dividing by 60 (35 min = 35/60 h, not 0.35 h).
• Compound measures:

Measure	Formula	Typical units
Speed	distance ÷ time	km/h, m/s
Density	mass ÷ volume	g/cm³, kg/m³
Pressure	force ÷ area (given in the exam)	N/m²

• km/h to m/s: multiply by 1000, divide by 3600 (or divide by 3.6).
• Reading scales: find the value of one small division first. If 5 divisions span 0 g to 100 g, each division is 20 g.
• Measuring angles: use the protractor scale that starts at 0° on the arm you have lined up, and give the answer to the nearest degree.
• Estimates: a door is about 2 m tall, a bag of sugar about 1 kg, a car about 4 m long.

Constructions (4.5)

Method: perpendicular bisector of AB

1. Compasses set to more than half AB.
2. Arcs above and below the line from A, then from B, radius unchanged.
3. Rule a line through the two intersections.

Method: angle bisector at B

1. Arc from B cutting both arms.
2. Equal arcs from those two points, crossing inside the angle.
3. Rule from B through the crossing.

Keep arcs visible. Scale drawings: real length = map length × scale factor, then convert units (100 000 cm = 1 km).

Circle facts (4.6)

Both tiers

• Tangent ⟂ radius at the point of contact.
• Tangents from one external point are equal in length.
• Perpendicular from the centre to a chord bisects the chord (and the converse).

Higher tier only

• Angle at centre = 2 × angle at circumference (same arc).
• Angle in a semicircle = 90°.
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral add up to 180°.
• Alternate segment theorem: angle between tangent and chord = angle in the alternate segment.
• Intersecting chords inside: AX × XB = CX × XD.
• Chords extended to meet outside at P: PA × PB = PC × PD.

Trigonometry and Pythagoras (4.8)

Tool	Use when	Formula
Pythagoras	right angle, two sides known, find third	a² + b² = c²
SOH CAH TOA	right angle, one side and one angle, or two sides and an angle wanted	sin = O/H, cos = A/H, tan = O/A
Sine rule (Higher)	a side-angle pair is known	a/sin A = b/sin B = c/sin C
Cosine rule (Higher)	two sides and included angle, or three sides	a² = b² + c² − 2bc cos A
Area (Higher)	two sides and included angle	½ab sin C

The sine rule, cosine rule and ½ab sin C are on the Higher formulae sheet; Pythagoras and SOH CAH TOA are not.

Method: angle from three sides (Higher)

1. Label the angle you want A and the side opposite it a.
2. cos A = (b² + c² − a²) ÷ 2bc.
3. A negative cosine means an obtuse angle, which the calculator gives directly.

Method: 3D problems (Higher)

1. Sketch the right-angled triangle you need, separately from the solid.
2. Find any missing base length first (often a diagonal, using Pythagoras).
3. For the angle between a line and a plane, use the triangle made by the line, its projection on the plane and the vertical.

Small worked reminders

Regular polygon, exterior angle 30°:  n = 360 ÷ 30 = 12 sides, interior 150°
Car travels 91 km in 1 h 45 min:      91 ÷ 1.75 = 52 km/h
Right-angled triangle, legs 9 and 12: hypotenuse = √(81 + 144) = 15
Angle with opposite 5, hypotenuse 8:  sin⁻¹(5/8) = 38.7°

Must-know distinctions

• Alternate vs allied: alternate angles are equal; allied angles add to 180°.
• Interior vs exterior angle: they add to 180° at each vertex; only the exterior angles always total 360°.
• Bearing of B from A vs of A from B: they differ by 180°.
• Elevation vs depression (Higher): both are measured from the horizontal, up or down; the angle of depression from A to B equals the angle of elevation from B to A.
• sin vs cos for obtuse angles (Higher): sin(180° − θ) = sin θ, but cos(180° − θ) = −cos θ. The sine rule can give two possible angles; the cosine rule cannot.
• Chord vs tangent vs radius: a chord joins two points on the circle; a tangent touches it once; a radius joins centre to circumference.

Quick self-test

1. A triangle has interior angles 48° and 67°. Find the exterior angle at the third vertex.
2. Find the sum of the interior angles of a 12-sided polygon.
3. Find the interior angle of a regular decagon.
4. The bearing of B from A is 312°. Find the bearing of A from B.
5. Convert 15 m/s to km/h.
6. A flight leaves at 10:50 pm and lands at 6:05 am the next day. How long is the flight?
7. A tangent from P touches a circle of radius 7 cm at T. OP = 25 cm, where O is the centre. Find PT.
8. The angle at the centre subtended by an arc is 98°. Find the angle at the circumference subtended by the same arc. (Higher)
9. One angle of a cyclic quadrilateral is 73°. Find the opposite angle. (Higher)
10. A right-angled triangle has hypotenuse 20 cm and an angle of 35°. Find the side opposite the 35° angle.
11. sin x = 0.6. Find both possible values of x between 0° and 180°. (Higher)
12. Find the area of a triangle with sides 5 cm and 8 cm and an included angle of 150°. (Higher)

Answers

1. 48° + 67° = 115° (exterior angle = sum of interior opposite angles)
2. (24 − 4) × 90° = 1800°
3. Exterior 360° ÷ 10 = 36°, interior 144°
4. 312° − 180° = 132°
5. 15 × 3.6 = 54 km/h
6. 1 h 10 min to midnight + 6 h 5 min = 7 h 15 min
7. √(25² − 7²) = √576 = 24 cm
8. 98° ÷ 2 = 49°
9. 180° − 73° = 107°
10. 20 sin 35° = 11.5 cm (3 s.f.)
11. 36.9° and 143.1° (1 d.p.)
12. ½ × 5 × 8 × sin 150° = 20 × 0.5 = 10 cm²

Where marks are usually lost

• Giving a reason as “Z angles” or “F angles” instead of “alternate angles are equal” or “corresponding angles are equal”.
• Writing only the final angle when the question says “give reasons for each stage”.
• Dividing 360° by n and calling it the interior angle.
• Converting 2 hours 35 minutes to 2.35 hours.
• Writing a bearing as 70° rather than 070°, or measuring from the wrong point (“from A” means stand at A).
• Rubbing out construction arcs, which removes the evidence the marks depend on.
• Using the circumference angle as the centre angle, or applying “angles in the same segment” to angles in different segments (Higher).
• Rounding intermediate lengths to 2 or 3 figures before using them again, so the final answer is outside the accepted range.
• Taking the acute answer from sin⁻¹ when the diagram shows an obtuse angle (Higher).
• In 3D, using a slant edge where the vertical height is needed (Higher).

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 4, Geometry and trigonometry, sections 4.1–4.8 (Foundation and Higher tier content).