Unit 3 · Topic 3: Geometry and Trigonometry
The Complete IB Mathematics: Applications and Interpretation Guide · IB Mathematics: Applications and InterpretationUnit 3 · Topic 3: Geometry and Trigonometry
- The Complete IB Mathematics: Applications and Interpretation Guide
- IB Mathematics: Applications and Interpretation
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. This topic develops measurement, trigonometric modeling, vectors, and networks. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Geometry and Trigonometry: choose, calculate, interpret” page.
- How IB Mathematics: Applications and Interpretation assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Applications and Interpretation; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a labeled triangle or three-dimensional diagram, a one-variable surface-area model with a feasible domain, a weighted network with route totals
- Key skills Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles, Sine Rule, Cosine Rule, Area of a Triangle, and Applications, The Circle: Arc Length and Sector Area, Perpendicular Bisectors and Voronoi Diagrams, Radians, the Unit Circle, Trigonometric Identities and Equations (HL), Geometric Transformations with Matrices (HL), Vectors: Lines, Kinematics, Scalar and Vector Products (HL), Graph Theory: Adjacency Matrices, Walks, Trees, and Cycle Algorithms (HL)
- How to study for Unit 3 This page turns a labeled triangle or three-dimensional diagram, a one-variable surface-area model with a feasible domain, a weighted network with route totals into one route: Draw the geometry or network and label every side, angle, unit, vertex, and edge weight before calculating..
- The organizing decision Translate the setting into a labeled diagram or network, then choose the geometric relation or optimization model that preserves its restrictions.
What IB Mathematics: Applications and Interpretation Unit 3 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles
bearing; cosine rule; weighted edge02Sine Rule, Cosine Rule, Area of a Triangle, and Applications
bearing; cosine rule; weighted edge03The Circle: Arc Length and Sector Area
bearing; cosine rule; weighted edge04Perpendicular Bisectors and Voronoi Diagrams
bearing; cosine rule; weighted edge05Radians, the Unit Circle, Trigonometric Identities and Equations (HL)
bearing; cosine rule; weighted edge06Geometric Transformations with Matrices (HL)
bearing; cosine rule; weighted edge07Vectors: Lines, Kinematics, Scalar and Vector Products (HL)
bearing; cosine rule; weighted edge08Graph Theory: Adjacency Matrices, Walks, Trees, and Cycle Algorithms (HL)
bearing; cosine rule; weighted edgeInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Geometry and Trigonometry: choose, calculate, interpret
Connect the published share to the unit model
This topic develops measurement, trigonometric modeling, vectors, and networks. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Geometry and Trigonometry: choose, calculate, interpret” page.
Vocabulary in use: bearing, radian, sine rule, cosine rule, surface area, vector, vertex, weighted edge. Each term is applied on these pages and collected again in the glossary with a checked page pointer on the “Geometry and Trigonometry: choose, calculate, interpret” page.
A strong geometry solution labels sides, angles, units, and network weights before choosing a relation, then checks feasibility after calculating. The mast, hiking, cylinder, and delivery situations in this topic are independently written illustrations.
The decision that organizes this unit
Define the system and choose the route before calculating
Translate the setting into a labeled diagram or network, then choose the geometric relation or optimization model that preserves its restrictions.
Draw the geometry or network and label every side, angle, unit, vertex, and edge weight before calculating.
Mechanism route and repair branches
Relationships to preserve
- Tangent links opposite and adjacent sides in a right triangle.
- The cosine rule links three sides with an included or opposite angle.
- A constrained optimization model uses one relation to remove a variable before testing a feasible minimum or maximum.
Representations to read
- a labeled triangle or three-dimensional diagram
- a one-variable surface-area model with a feasible domain
- a weighted network with route totals
Branches to reject
- using the wrong angle mode or leaving instrument height assumptions unstated
- rounding an intermediate value before the final geometric result
- choosing one route without comparing every feasible total
| Key concept | Why it's hard | What scores |
|---|---|---|
| bearing | Distinguishing bearing from radian requires their definitions and units. | Define the quantity measured by bearing and attach its units before calculation. |
| sine rule | A calculator screen does not show why sine rule is the suitable model feature. | Record the relation used for sine rule and one identifying intermediate value. |
| vector | A computed vector value still needs a domain and scale check. | Check the restrictions, precision, and contextual meaning of the vector result. |
How IB Mathematics: Applications and Interpretation assesses Topic 3: Geometry and Trigonometry
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles | bearing; cosine rule; weighted edge | using the wrong angle mode or leaving instrument height assumptions unstated |
| Sine Rule, Cosine Rule, Area of a Triangle, and Applications | bearing; cosine rule; weighted edge | rounding an intermediate value before the final geometric result |
| The Circle: Arc Length and Sector Area | bearing; cosine rule; weighted edge | choosing one route without comparing every feasible total |
| Perpendicular Bisectors and Voronoi Diagrams | bearing; cosine rule; weighted edge | using the wrong angle mode or leaving instrument height assumptions unstated |
| Radians, the Unit Circle, Trigonometric Identities and Equations (HL) | bearing; cosine rule; weighted edge | rounding an intermediate value before the final geometric result |
| Geometric Transformations with Matrices (HL) | bearing; cosine rule; weighted edge | choosing one route without comparing every feasible total |
| Vectors: Lines, Kinematics, Scalar and Vector Products (HL) | bearing; cosine rule; weighted edge | using the wrong angle mode or leaving instrument height assumptions unstated |
| Graph Theory: Adjacency Matrices, Walks, Trees, and Cycle Algorithms (HL) | bearing; cosine rule; weighted edge | rounding an intermediate value before the final geometric result |
Resolve a right-triangle survey
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Resolve a mast survey
From a point 24 m from the base of a vertical mast on level ground, the angle of elevation to its top is 31°. The instrument height is negligible.
Calculate the height of the mast.
Show your working.
Give your answer to three significant figures.
A. tan 31° = h ÷ 24, so h = 24 tan 31° = 14.4 m.
✅ Correct! 😊 Correct. The mast height is opposite the angle and the ground distance is adjacent, so tangent gives the required height.
B. tan 31° = 24 ÷ h, so h = 24 ÷ tan 31° = 39.9 m.
❌ Not quite 😭 This swaps the opposite and adjacent sides. The 24 m ground distance is adjacent to the angle, not opposite it.
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C. sin 31° = h ÷ 24, so h = 12.4 m.
❌ Not quite 😭 Sine requires the hypotenuse, but 24 m is the horizontal adjacent side. The chosen ratio does not match the diagram.
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D. h = 24 tan 31 = −10.6 m.
❌ Not quite 😭 This evaluates 31 in radian mode. A negative height is also incompatible with the survey setting.
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Full explanation
- Step 1In the right triangle, label h opposite the 31° angle and 24 m adjacent to it.
- Step 2Use tangent because tan θ = opposite ÷ adjacent: tan 31° = h ÷ 24.
- Step 3In degree mode, h = 24 tan 31° = 14.42 m, which rounds to 14.4 m.
- Step 4The result assumes level ground and negligible instrument height, as stated.
Practice 2: Compare weighted routes
A delivery network has three feasible routes from P to S. Route P–Q–S has edge weights 5 and 8; P–R–S has weights 7 and 4; P–Q–R–S has weights 5, 1 and 4.
Determine every route with minimum total weight.
Show all route totals.
A. P–Q–S is minimum because it uses only two edges.
❌ Not quite 😭 The number of edges does not determine total weight. This route totals 13, which is larger than the alternatives.
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B. P–R–S alone is minimum with total weight 11.
❌ Not quite 😭 Its total is 11, but the three-edge route totals 10. Stopping after one comparison misses the smaller route.
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C. P–Q–R–S is minimum with total weight 10.
✅ Correct! 😊 Correct. The three route totals are 13, 11 and 10, so this complete route has the unique minimum.
D. P–Q–R–S is minimum with total weight 6.
❌ Not quite 😭 This omits the final R–S edge of weight 4. A route total must include every edge from P to S.
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Full explanation
- Step 1List every feasible complete path before choosing a minimum.
- Step 2The totals are P–Q–S: 5 + 8 = 13; P–R–S: 7 + 4 = 11; P–Q–R–S: 5 + 1 + 4 = 10.
- Step 3Compare 13, 11 and 10; the unique minimum total is 10.
- Step 4Therefore, P–Q–R–S is the only minimum-weight route.
Key terms for Unit 3: Topic 3: Geometry and Trigonometry
Models, uses, and boundaries
- Relate Angle and Side Lengths
- Tangent links opposite and adjacent sides to an angle; label those sides before substitution. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Resolve a Right-Triangle Survey
- Use tan 31° = h ÷ 24 to obtain h = 24 tan 31° = 14.42 m; check degree mode, level ground, and the negligible instrument-height assumption. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Use the Cosine Rule
- Evaluate d² = 7.2² + 9.5² − 2(7.2)(9.5)cos 64° before taking the square root; check that d = 9.062 km is feasible between the difference and sum of the two legs. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Optimize a Cylinder
- Substitute h = 500 ÷ (πr²) into surface area to form S(r) = 2πr² + 1000 ÷ r; solve S′(r) = 0 and retain the positive feasible radius r = 4.30 cm. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Compare Weighted Routes
- Total A–B–D as 11, A–C–D as 9, and A–B–C–D as 9; compare all feasible totals and recognize that two routes share the minimum weight. Use this relation when the named quantities and conditions in the question match the calculation described here.
IB Mathematics: Applications and Interpretation Unit 3 FAQ
How much of IB Mathematics: Applications and Interpretation does Unit 3 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 3: Geometry and Trigonometry problem?
Draw the geometry or network and label every side, angle, unit, vertex, and edge weight before calculating.
Which relationships should I preserve?
Tangent links opposite and adjacent sides in a right triangle. The cosine rule links three sides with an included or opposite angle. A constrained optimization model uses one relation to remove a variable before testing a feasible minimum or maximum.
Which representations should I practice?
Practice moving among a labeled triangle or three-dimensional diagram, a one-variable surface-area model with a feasible domain, a weighted network with route totals.
What error should I check before submitting an answer?
Check for using the wrong angle mode or leaving instrument height assumptions unstated; rounding an intermediate value before the final geometric result; choosing one route without comparing every feasible total.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Resolve a right-triangle survey
- Use the cosine rule before rounding
- Check and answer: Resolve a right-triangle survey · Use the cosine rule before rounding
- Minimize material for a cylinder
- Compare routes in a weighted network
- Check the optimization and network answers
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Applications and Interpretation unit guides
IB Mathematics: Applications and Interpretation Exam Guide & Review
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International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Applications and Interpretation Unit 3
Start with the organizing decision
Before solving, restate the decision in operational terms: Translate the setting into a labeled diagram or network, then choose the geometric relation or optimization model that preserves its restrictions.. Your first written move should be to Draw the geometry or network and label every side, angle, unit, vertex, and edge weight before calculating..
Practice the same idea in several representations
Rotate through a labeled triangle or three-dimensional diagram, a one-variable surface-area model with a feasible domain, a weighted network with route totals. Use each representation to practice Three-Dimensional Geometry: Distance, Midpoint, Volume, Surface Area, and Angles, Sine Rule, Cosine Rule, Area of a Triangle, and Applications, The Circle: Arc Length and Sector Area, Perpendicular Bisectors and Voronoi Diagrams, Radians, the Unit Circle, Trigonometric Identities and Equations (HL), Geometric Transformations with Matrices (HL), Vectors: Lines, Kinematics, Scalar and Vector Products (HL), Graph Theory: Adjacency Matrices, Walks, Trees, and Cycle Algorithms (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using the wrong angle mode or leaving instrument height assumptions unstated; rounding an intermediate value before the final geometric result; choosing one route without comparing every feasible total. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Applications and Interpretation to see every unit and plan your revision.