IB DP Mathematics: Analysis and Approaches – 3D geometry, triangle trigonometry and radian measure Revision Notes

For full explanations and longer worked examples, read the study guide first. These notes condense the 3D geometry, triangle trigonometry and radian measure unit of IB Diploma Programme Mathematics: Analysis and Approaches. They are aligned to the IB Mathematics: analysis and approaches guide, first assessment 2021, and cover syllabus sections 3.1–3.4, which are SL content examined at both SL and HL. They follow the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so they apply to the May and November 2026, 2027 and 2028 sessions.

Test yourself afterwards with the practice questions. The IB DP Maths AA course hub and the printable syllabus checklist show where this unit sits in the course.

Definitions

• Radian: the angle at the centre of a circle subtended by an arc equal in length to the radius. 2π rad = 360°, so π rad = 180°.
• Angle of elevation: measured upwards from the horizontal to the line of sight.
• Angle of depression: measured downwards from the horizontal to the line of sight.
• Bearing: a three-figure angle measured clockwise from north (for example 007°, 245°).
• Angle between a line and a plane: the angle between the line and its projection onto the plane.
• Right pyramid / right cone: the apex is directly above the centre of the base.
• Slant height l of a cone: the distance from the apex to the edge of the base; l² = r² + h².

Formulas

The guide states that all formulae required for the course are in the mathematics formula booklet. Know them well enough to choose the right one fast.

Topic	Formula
3D distance	d = √((x₁ − x₂)² + (y₁ − y₂)² + (z₁ − z₂)²)
3D midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
Right pyramid	V = (1/3)Ah, A = base area
Right cone	V = (1/3)πr²h; curved surface area = πrl
Sphere	V = (4/3)πr³; S = 4πr²
Hemisphere	V = (2/3)πr³; curved 2πr²; total (with flat face) 3πr²
Sine rule	a/sin A = b/sin B = c/sin C
Cosine rule	c² = a² + b² − 2ab cos C
Cosine rule (angle)	cos C = (a² + b² − c²)/(2ab)
Triangle area	½ab sin C
Degrees ↔ radians	× π/180 to radians; × 180/π to degrees
Arc length	l = rθ (θ in radians)
Sector area	A = ½r²θ (θ in radians)

Method in steps

Choosing a triangle rule

Right angle in the triangle?         → SOH CAH TOA or Pythagoras
Know a side AND its opposite angle?  → sine rule
Know two sides and included angle?   → cosine rule for the third side
Know all three sides?                → cosine rule for an angle
Want the area?                       → ½ab sin C (angle between a and b)

Angle between a line and a plane (3D solid)

1. Mark the point where the line meets the plane.
2. From another point on the line, drop a perpendicular to the plane.
3. Join the foot of the perpendicular to the meeting point. This is the projection.
4. You now have a right-angled triangle. Use tan, sin or cos.

Combined solid

1. Split into named solids (cone, hemisphere, pyramid, sphere).
2. Volume: add the parts.
3. Surface area: list each outside face. Leave out faces that are joined.
4. Give exact answers in π on calculator-free questions.

Elevation from two observation points

1. Sketch the tower and both points on one horizontal line.
2. At the nearer point, the angle inside the triangle is 180° minus its angle of elevation.
3. The angle at the top is the difference between the two angles of elevation.
4. Use the sine rule in the non-right-angled triangle to find the sloping distance from the nearer point.
5. Finish in the right-angled triangle: height = sloping distance × sin(nearer elevation).

Segment problems

1. Find the sector area ½r²θ with θ in radians.
2. Find the triangle area ½r² sin θ.
3. Segment area = sector − triangle.
4. For the segment perimeter, find the chord with the cosine rule and add the arc rθ.

Bearings problem

1. Draw a north line at every point used.
2. Mark each bearing clockwise from north.
3. Use back bearing = bearing ± 180° to find angles inside the triangle.
4. Solve the triangle, then convert back to a bearing from north.

Small worked reminders

• A(0, 3, −1), B(4, 5, 3): AB = √(16 + 4 + 16) = √36 = 6; midpoint (2, 4, 1).
• 135° = 135 × π/180 = 3π/4; 1 rad = 180/π ≈ 57.3°.
• Sector with r = 5 and θ = 1.2: arc = 6, area = ½ × 25 × 1.2 = 15.
• Cone with r = 5, h = 12: l = 13, curved surface = 65π, V = 100π.
• Sides 7, 9, 12: cos C = (49 + 81 − 144)/126 = −1/9, so C = 96.4°, which is obtuse.
• Cliff 80 m high, depression 14° to a boat: horizontal distance = 80/tan 14° = 321 m.
• Segment area = sector − triangle = ½r²θ − ½r² sin θ.
• Chord for r = 9, θ = 1.8: AB² = 81 + 81 − 162 cos 1.8, so AB = 14.1; with arc 16.2 the segment perimeter is 30.3.
• Rectangular base 8 by 6 with apex 12 above the centre: half-diagonal 5, edge 13, angle with base tan⁻¹(12/5) = 67.4°.

Must-know distinctions

Pair	The difference
Degrees vs radians	Arc and sector formulas need radians. Exam papers assume radians unless told otherwise (guide).
Arc length vs sector perimeter	Perimeter = rθ + 2r. The arc alone is rθ.
Height vs slant height	V = (1/3)πr²h uses h; curved area πrl uses l.
Sphere vs hemisphere surface	4πr² vs 2πr² curved, 3πr² with the flat face.
Elevation vs depression	Both measured from the horizontal; up vs down. They are equal for the same two points.
Sine rule vs cosine rule	Sine rule needs a matching side–angle pair; cosine rule does not.
Sine rule for an angle	Gives an acute answer. Safe if the angle is opposite the shorter known side. The ambiguous case is in section 3.5, not here.
SL vs HL 3D questions	The guide says SL examinations set only right-angled trigonometry on 3D shapes.

Calculator-free vs calculator allowed

Paper 1 (SL and HL) allows no technology. Paper 2 requires it, as does HL Paper 3. On Paper 1 expect exact answers: surds, multiples of π, fractions. On Paper 2, give answers to 3 significant figures unless told otherwise, and check your angle mode.

Quick self-test

1. Find the distance between (1, 2, −2) and (3, −4, 1).
2. Find the midpoint of (−3, 5, 2) and (7, 1, −6).
3. Write 225° in radians as a multiple of π.
4. Write 2 radians in degrees, to 3 significant figures.
5. A sector has radius 10 cm and angle 0.7 rad. Find the arc length.
6. A sector has radius 4 cm and angle 3π/8. Find its exact area.
7. Find the exact volume of a sphere of radius 3 cm.
8. Find the exact total surface area of a solid hemisphere of radius 5 cm.
9. A right cone has radius 5 cm and height 12 cm. Find its exact curved surface area and volume.
10. Find the area of a triangle with sides 7 cm and 10 cm and included angle 30°, given sin 30° = ½.
11. A triangle has sides 3, 5 and 7. Find its largest angle (calculator allowed).
12. A sector has radius 8 cm and arc length 12 cm. Find the angle in radians.

Answers

1. √(2² + 6² + 3²) = √49 = 7
2. (2, 3, −2)
3. 225 × π/180 = 5π/4
4. 2 × 180/π = 114.59… = 115°
5. l = 10 × 0.7 = 7 cm
6. ½ × 16 × 3π/8 = 3π cm²
7. (4/3)π × 27 = 36π cm³
8. 3π × 25 = 75π cm²
9. l = √(25 + 144) = 13; curved area = π × 5 × 13 = 65π cm²; V = (1/3)π × 25 × 12 = 100π cm³
10. ½ × 7 × 10 × ½ = 17.5 cm²
11. cos C = (9 + 25 − 49)/30 = −½, so C = 120°
12. θ = 12/8 = 1.5 rad

Where marks are usually lost

• Using l = rθ or ½r²θ with θ in degrees, or leaving the GDC in degree mode on a radian question.
• Giving a sector’s perimeter as rθ and forgetting the two radii.
• Using vertical height in πrl, or slant height in (1/3)πr²h.
• Counting a joined face in the surface area of a combined solid.
• Measuring the angle between a line and a plane against an edge of the plane instead of the projection.
• Bearings not written with three figures, or measured anticlockwise.
• Rounding a side to 3 s.f. and carrying it into the next step, which shifts the final answer.
• Using the sine rule for the largest angle of a triangle when it may be obtuse.
• Decimal answers on Paper 1 questions that ask for exact values.
• No labelled diagram, so the angle used in the working is not clear to the examiner.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: analysis and approaches guide, first assessment 2021 (published February 2019, updated November 2020).