# Geometry and Trigonometry

> IB Mathematics Applications and Interpretation SL · IB AI SL Unit 3
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-overview/
> Weight: 12-14% of overall IB AI SL exam

This unit covers practical geometric and trigonometric tools for 3D measurement, triangle problem solving, periodic modelling, and spatial analysis, all heavily tested in both IB AI SL papers.

**Prerequisites:** Foundational algebra and function notation; Basic angle properties of triangles and straight lines

## Learning objectives

- Calculate volume and surface area for common 3D solids and composite solids in real-world problems
- Apply trigonometric rules to right and non-right triangles to solve measurement and navigation problems
- Model periodic real-world phenomena (tides, temperature, motion) using transformed sinusoidal functions
- Construct and use Voronoi diagrams to solve practical closest-site spatial problems

## Unit at a Glance

This unit builds from basic 2D and 3D measurement up to applied trigonometry and two AI SL-specific core topics: sinusoidal modelling and Voronoi diagrams. Skills build incrementally, with early trigonometry concepts reused in later applications and modelling work.

Approximately half of the unit's exam marks come from applied problems, so we focus on connecting formulas to real-world contexts throughout the sub-topics, which aligns with AI SL's assessment focus.

All sub-topics in this unit, ordered for sequential learning:
- [Volume and surface area of 3D solids](https://www.owlsprep.com/study/ib-math-ai-sl-u3-volume-and-surface-area-of/) — Calculate volume and surface area for common 3D solids and composite solids.
- [Right-angled triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-angled-triangle-trigonometry/) — Use SOHCAHTOA and Pythagoras' theorem to solve for unknown sides and angles.
- [Arc length and sector area](https://www.owlsprep.com/study/ib-math-ai-sl-u3-arc-length-and-sector-area/) — Find the length of an arc and the area of a sector of a circle, working in degrees.
- [Sine rule, cosine rule for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sine-rule-cosine-rule-for/) — Apply the sine and cosine rules to find unknown sides and angles in any triangle.
- [Bearings and trigonometric applications](https://www.owlsprep.com/study/ib-math-ai-sl-u3-bearings-and-trigonometric-applications/) — Solve real-world navigation problems using bearings and combined trigonometric rules.
- [Sinusoidal functions: amplitude, period, translation](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sinusoidal-functions-amplitude-period-translation/) — Model periodic phenomena with transformed sine and cosine functions.
- [Voronoi diagrams and closest site problems](https://www.owlsprep.com/study/ib-math-ai-sl-u3-voronoi-diagrams-and-closest-site/) — Construct and interpret Voronoi diagrams for closest facility spatial problems.
- [Triangle area formula for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-triangle-area-formula-for-non/) — Calculate the area of any triangle using the $\frac{1}{2}ab \sin C$ formula.

## Common pitfalls

- **Wrong:** Forgetting to convert bearings to the internal angle of a triangle before applying trig rules
  - Why it fails: Bearings are measured clockwise from north, so they do not directly give the required triangle internal angle
  - Correct: Draw a north line at each vertex and use parallel line/angle sum rules to find the internal angle
- **Wrong:** Confusing the formula for the period of a sinusoidal function $y = \sin(kx)$
  - Why it fails: Many students incorrectly use $period = k/2\pi$ instead of the correct relationship
  - Correct: Remember $period = 2\pi/|k|$ for sinusoids in the standard transformed form
- **Wrong:** Double-counting overlapping faces when calculating surface area of composite 3D solids
  - Why it fails: When two solids are joined, the overlapping faces are no longer on the surface
  - Correct: Subtract twice the area of the overlapping face from the sum of individual surface areas

## Cheatsheet

| Concept | Key Formula / Rule |
| --- | --- |
| Volume of a sphere | $V = \frac{4}{3}\pi r^3$ |
| Surface area of a sphere | $SA = 4\pi r^2$ |
| Sine Rule | $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ |
| Cosine Rule (for side $a$) | $a^2 = b^2 + c^2 - 2bc \cos A$ |
| Area of any triangle | $Area = \frac{1}{2}ab \sin C$ |
| Period of $y = \sin(kx)$ | $Period = \frac{2\pi}{\|k\|}$ |
| Voronoi edge property | All points on an edge are equidistant from the two adjacent sites |

## What's next

Start with the first sub-topic of this unit below to build your foundational 3D measurement skills, then work through each sub-topic in order to progress to more complex applications and modelling. After completing all sub-topics in this unit, you can move on to the next unit overview for Statistics and Probability.

- [Volume and surface area of 3D solids](https://www.owlsprep.com/study/ib-math-ai-sl-u3-volume-and-surface-area-of/)
- [Unit 4: Statistics and Probability Overview](https://www.owlsprep.com/study/ib-math-ai-sl-u4-overview/)
- [Right-angled triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-angled-triangle-trigonometry/)

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From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u3-overview/