IB DP Mathematics: Applications and Interpretation – 3D geometry, triangle trigonometry, bearings and sectors Revision Notes

For full explanations and worked examples, read the study guide first.

These revision notes cover 3D geometry, triangle trigonometry, bearings and sectors for IB Diploma Programme Mathematics: Applications and Interpretation. They are aligned to the IB Mathematics: applications and interpretation guide, first assessment 2021, syllabus sections 3.1 to 3.4, which are common content for 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 AI course hub, the printable syllabus checklist and the geometry and trigonometry strand overview put the unit in context.

Scope in one glance

• 3.1 3D distance and midpoint; volume and surface area of right pyramids, right cones, spheres, hemispheres and combinations; angle between two intersecting lines, or between a line and a plane.
• 3.2 SOH CAH TOA; sine rule; cosine rule; area = (1/2)ab sin C.
• 3.3 Applications, including Pythagoras, elevation, depression and bearings; labelled diagrams from words.
• 3.4 Arc length and sector area, in degrees.

Limits set by the guide: SL exams only set right-angled trigonometry on 3D shapes; no ambiguous case of the sine rule; radians are not required at SL (radians are HL only, AHL 3.7). The guide says all formulae required for the course are in the formula booklet.

Formulas

Result	Formula
3D distance	d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
3D midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
Right pyramid volume	V = (1/3)Ah, A = base area
Right cone	V = (1/3)πr²h; curved surface = πrl; l = √(r² + h²)
Sphere	V = (4/3)πr³; S = 4πr²
Hemisphere	V = (2/3)πr³; curved surface = 2πr² (+ πr² for the flat face)
Sine rule	a/sin A = b/sin B = c/sin C
Cosine rule	c² = a² + b² − 2ab cos C; cos C = (a² + b² − c²)/(2ab)
Triangle area	(1/2)ab sin C
Arc length	(θ/360) × 2πr
Sector area	(θ/360) × πr²

Method in steps

Angle between a line and a plane

1. Drop a perpendicular from a point on the line to the plane.
2. Join the foot to where the line meets the plane (the projection).
3. Solve the right-angled triangle: line = hypotenuse, perpendicular = opposite.

Surface area of a combined solid

1. Sketch it and list every face.
2. Cross out faces that are joined together.
3. Add the remaining areas. Use slant height for cones and triangular faces.

Bearings problem

1. Draw a north line at every point.
2. Mark each bearing clockwise from north.
3. Find the interior angle at the turning point: back bearing = forward bearing ± 180°, then subtract.
4. Cosine rule for the distance; sine rule for an angle; add or subtract from a known bearing.
5. Give the bearing as three figures.

Choosing a rule

• Two angles and a side: sine rule.
• Two sides and the included angle: cosine rule.
• Three sides: cosine rule, rearranged for cos.
• Right angle present: SOH CAH TOA or Pythagoras. Faster and safer.

Small worked reminders

Cuboid 8 × 6 × 5 (height 5). Base diagonal = √(8² + 6²) = 10. Space diagonal = √(10² + 5²) = √125 = 11.2. Angle between space diagonal and base = tan⁻¹(5/10) = 26.6°.

Elevation. A kite string makes 40° with the ground. The angle of depression from the kite to the person holding the string is also 40°, because the two horizontals are parallel.

Reverse sector. Arc length 20 cm, radius 9 cm: θ = 20 × 360/(2π × 9) = 127.3°.

Obtuse from cosine. Sides 7, 9, 12: cos C = −14/126, so C = 96.4°. A negative cosine always means an obtuse angle.

Combined solid. Hemisphere radius 6 cm with a cone of height 8 cm on its flat face: l = 10, so V = 144π + 96π = 240π = 754 cm³ and surface = 2π(36) + π(6)(10) = 132π = 415 cm². The joined circles are not counted.

Pyramid angles. Square base 10 m, height 12 m. Sloping edge to base: tan⁻¹(12/(5√2)) = 59.5°. Apex-to-edge-midpoint line to base: tan⁻¹(12/5) = 67.4°. Same pyramid, two different projections, two different answers.

Elevation from two points. Angles 32° at A and 51° at B, with AB = 40 m towards the tower. The angle at the top is 51° − 32° = 19°. Sine rule gives the slant distance from B, then multiply by sin 51° for the height (50.6 m).

Bearing turn. Out on 040°, then on 130°. Back bearing at the turn = 220°, so the interior angle = 220° − 130° = 90°. Always check whether the turn gives a right angle before reaching for the cosine rule.

Segment. Radius 10 m, angle 80°: sector 69.81 m² minus triangle (1/2)(10²) sin 80° = 49.24 m² gives 20.6 m².

GDC checklist

The guide lists technology as required on every AI paper. Use it, but show the set-up.

• Degree mode, checked at the start of every paper.
• Store unrounded values in memory; round only the final answer.
• Use the solver for reverse problems (find r from a volume, θ from an arc length).
• Write the formula with numbers substituted before pressing enter. That line carries the method mark.
• Round to 3 significant figures unless told otherwise. Give bearings as three figures.

Must-know distinctions

Pair	Difference
Vertical height vs slant height	Vertical height goes in V = (1/3)πr²h; slant height l goes in πrl
Elevation vs depression	Both measured from the horizontal; elevation looks up, depression looks down
Arc length vs sector area	Arc uses 2πr (length); sector uses πr² (area)
Sector perimeter vs arc length	Perimeter = arc + 2r
Sector vs segment	Segment = sector − triangle, triangle area = (1/2)r² sin θ
Line-plane angle in a pyramid	Sloping edge: projection is half a diagonal. Apex to edge midpoint: projection is half a side
Sphere vs hemisphere surface	Sphere 4πr²; hemisphere curved 2πr², total 3πr² if the flat face shows

Quick self-test

1. Find the distance between (1, 2, 3) and (4, 6, 15).
2. Find the midpoint of (−2, 5, 1) and (6, −3, 7).
3. Find the volume of a sphere of radius 3 cm.
4. A right cone has radius 7 cm and height 24 cm. Find its slant height and curved surface area.
5. A right pyramid has a rectangular base 6 cm by 4 cm and height 9 cm. Find its volume.
6. A right-angled triangle has hypotenuse 20 cm and an angle of 35°. Find the side opposite the 35° angle.
7. In triangle ABC, A = 50°, B = 60° and a = 8 cm. Find b.
8. In triangle ABC, a = 5 cm, b = 8 cm and C = 60°. Find c.
9. Find the area of the triangle in question 8.
10. A sector has radius 6 cm and angle 120°. Find its arc length and area.
11. From the top of a 60 m cliff, the angle of depression of a buoy is 25°. How far is the buoy from the foot of the cliff?
12. The bearing of B from A is 062°. Find the bearing of A from B.

Answers

1. √(3² + 4² + 12²) = √169 = 13
2. (2, 1, 4)
3. (4/3)π(27) = 36π = 113 cm³
4. l = √(49 + 576) = 25 cm; curved surface = π(7)(25) = 175π = 550 cm²
5. (1/3)(24)(9) = 72 cm³
6. 20 sin 35° = 11.5 cm
7. b = 8 sin 60°/sin 50° = 9.04 cm
8. c² = 25 + 64 − 80 cos 60° = 49, so c = 7 cm
9. (1/2)(5)(8) sin 60° = 17.3 cm²
10. arc = 4π = 12.6 cm; area = 12π = 37.7 cm²
11. 60/tan 25° = 129 m
12. 062° + 180° = 242°

Where marks are usually lost

• Using vertical height where slant height is needed (or the reverse) in cone and pyramid surface areas.
• Including a joined face in the surface area of a combined solid.
• Using a whole diagonal instead of half a diagonal as the projection of a pyramid’s sloping edge.
• Applying the sine rule to a two-sides-and-included-angle triangle.
• Rounding a length early, then using it in the next step, so the final answer is wrong in the third figure.
• Missing the north line at the turning point of a bearings problem, so the interior angle is wrong.
• Giving a bearing as “77°” instead of three figures, 077°, or measuring it anticlockwise.
• Leaving out the two radii when a sector perimeter is asked for.
• Answering in radians because the calculator was in radian mode.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: applications and interpretation guide, first assessment 2021 – syllabus sections SL 3.1, SL 3.2, SL 3.3 and SL 3.4. See also the AI subject guide.