IB DP Mathematics: Applications and Interpretation – Geometry and Trigonometry Strand

This guide covers the Geometry and Trigonometry strand, which carries 18 hours at SL and jumps to 46 at HL – the largest proportional SL-to-HL increase of the five strands – for IB Diploma Programme Mathematics: Applications and Interpretation, first assessment 2021. The full syllabus guide gives the hours; the additional higher level content behind that jump is radians and circular functions, matrix transformations, vectors and graph theory (AHL 3.7-3.16), alongside Statistics and Probability, this course’s other large strand.

Where this fits in the syllabus

Unlike Analysis and Approaches’ more abstract, proof-oriented treatment of the same strand name, this course frames geometry and trigonometry around real-world spatial problems – navigation, design, and physical measurement contexts – and every external paper allows technology throughout, so questions routinely expect a graphical display calculator to be used, not avoided. Revision should reflect that applied, technology-permitted framing rather than treating this strand as pure abstract geometry.

Syllabus coverage

IB DP MATHEMATICS: APPLICATIONS AND INTERPRETATION – GEOMETRY AND TRIGONOMETRY

• Geometry – properties of shapes in two and three dimensions, surface area and volume of compound solids, and geometric reasoning applied to real contexts such as construction or design
• Trigonometry – right-angled and non-right-angled triangle trigonometry (sine rule, cosine rule, area of a triangle formula), angles of elevation and depression, and bearings – the classic navigation-style application this course favours
• Radians and circular functions (HL only, AHL 3.7-3.8) – the definition of a radian, and the definitions of cosine and sine as circular functions
• Matrix transformations (HL only, AHL 3.9) – geometric transformations using matrices
• Vectors (HL only, AHL 3.10-3.13) – position and displacement vectors, the vector equation of a line, vector applications to kinematics (problems of motion) and intersection, and the scalar and vector products
• Graph theory (HL only, AHL 3.14-3.16) – graphs, vertices and edges; adjacency matrices and walks; tree and cycle algorithms

Sine rule vs cosine rule – which to use

Given	Use
Two sides and the angle between them (SAS), or three sides (SSS)	Cosine rule
Two angles and a side, or two sides and a non-included angle	Sine rule

Always sketch the triangle first and label knowns before choosing a rule – misapplying the sine rule to an SAS triangle is one of the most common technique errors in this strand.

Worked example: bearings and the cosine rule

A ship sails from port A on a bearing of $070°$ for 40 km to point B, then changes course to a bearing of $150°$ and sails a further 25 km to point C. Find the distance AC. The bearing changes from $070°$ to $150°$, so the interior angle at B – found correctly from the bearings, allowing for the reverse bearing at B – is the step most students lose marks on, and is worth establishing carefully with a diagram before any calculation begins. Once the angle at B is known, the cosine rule applies directly: $AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(\angle ABC) = 40^2 + 25^2 - 2(40)(25)\cos(\angle ABC)$, and taking the square root gives AC in km. The mathematics itself is routine once the angle is correctly identified – the applied skill this course specifically tests is converting bearings into an interior angle correctly, which is why bearings problems reward a clear diagram before any calculation begins.

Surface area and volume of compound solids

Because this course frames geometry around real-world design and construction contexts, questions frequently present a compound solid – for example a cylinder topped with a hemisphere, or a pyramid combined with a cuboid – rather than a single standard shape. The reliable method is to split the compound solid into its recognisable component shapes, calculate each one’s surface area or volume separately using the standard formulae, then add or subtract components as the shape requires, subtracting for instance where one solid has a cavity removed from another. Sketching the solid and labelling which faces are “internal,” and therefore excluded from a surface area total, before calculating is the single most effective way to avoid the most common error in this content: including or excluding the wrong faces from a compound surface area.

How to approach it

Practise converting a bearing into an interior triangle angle correctly before attempting the cosine or sine rule calculation, since this conversion, not the trigonometric formula itself, is where most marks are actually lost. Memorise which rule applies to SAS, SSS, ASA and SSA triangle information, and sketch the triangle with knowns labelled before choosing. Split at least three different compound-solid shapes into components and practise both surface area and volume calculations for each, labelling internal faces explicitly. For HL, practise setting two lines’ parametric vector equations equal to test for intersection, rather than only comparing direction vectors. Because every paper in this course permits technology throughout, build the habit of using a graphical display calculator to sanity-check an algebraic or trigonometric answer, since this kind of verification is both permitted and expected.

Exam traps

Converting a bearing into an interior triangle angle incorrectly, especially when the bearing at the vertex is a “reverse” bearing (add or subtract $180°$) rather than the forward one. Choosing the sine rule for an SAS or SSS triangle where the cosine rule is required. Forgetting that HL vector questions on lines and intersection require setting the parametric forms of two lines equal and solving simultaneously, not just comparing direction vectors alone. Not using the graphical display calculator to sanity-check an answer, when every paper in this course permits technology throughout.

Official syllabus

International Baccalaureate Organization, Diploma Programme Subject Brief – Mathematics: Applications and Interpretation, first assessment 2021 – the same source already cited by the full syllabus guide and the revision notes already on the site. Verified 2026-09-06.