Geometry and Trigonometry (AI SL) — IB Math AI SL AI SL Study Guide

For: IB Math AI SL candidates sitting IB Math: Applications & Interpretation SL.

Covers: all core Geometry and Trigonometry subtopics for IB AI SL: Voronoi diagrams, right and non-right angle trigonometry, three-figure bearings, and surface area/volume calculation for 2D and 3D shapes.

You should already know: IGCSE / pre-DP math.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AI SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Is Geometry and Trigonometry (AI SL)?

Geometry and Trigonometry for IB Math AI SL is a practical, context-focused topic that teaches you to solve real-world spatial problems, from mapping service areas for businesses to calculating distances between ships at sea. It is weighted at 20-25% of your final exam grade, appearing in both Paper 1 (no calculator) and Paper 2 (calculator allowed), often combined with statistics, modeling, or financial application questions. Unlike pure math syllabi, AI SL prioritizes application over abstract proof, so all rules you learn will be tied to a tangible use case.

2. Voronoi diagrams

A Voronoi diagram is a spatial partitioning tool that divides a plane into regions (called cells) based on distance to fixed points called sites. Every point inside a cell is closer to its associated site than any other site. Key terms you must know:

• Site: Fixed reference point (e.g., a coffee shop location)
• Cell: Region of points closest to a single site
• Edge: Boundary between two cells, equal to the perpendicular bisector of the line segment joining the two adjacent sites
• Vertex: Intersection of three or more edges, equidistant from three or more sites

A common exam question asks for the largest empty circle, the biggest circle you can draw that does not contain any site: its center is always a Voronoi vertex, and its radius is the distance from that vertex to the nearest site.

Worked Example

You have three library sites at coordinates A(0,0), B(4,0), and C(2,3) on a 1km grid. Find the perpendicular bisector of A and B, and identify the maximum radius of a new library that will not be closer than any existing site to any local resident.

1. The midpoint of AB is (2,0), and AB is horizontal so its perpendicular bisector is the vertical line $x=2$.
2. The only Voronoi vertex is at (2, 1.08), equidistant from A, B, and C. The distance from this vertex to A is $\sqrt{(2{)}^2+(1.08{)}^2}\approx 2.27$ km, which is the maximum allowed radius for the new library’s service area.

3. Right-angle and non-right-angle trig

For right-angled triangles, you will use the standard SOHCAHTOA rules and Pythagoras’ theorem: $$a^2+b^2=c^2\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$$

For non-right-angled triangles, you have three core rules, derived from combining right-angle trig with Pythagoras’ theorem:

1. Sine Rule: For triangle with sides $a,b,c$ opposite angles $A,B,C$ respectively: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ Use when you have 2 angles + 1 side, or 2 sides + 1 non-included angle. Watch for the ambiguous case: if you calculate an angle from the sine rule, there may be 2 valid solutions (one acute, one obtuse) that fit the problem constraints.
2. Cosine Rule: $$a^2=b^2+c^2-2bc\cos A$$ Use when you have 2 sides + 1 included angle, or all 3 sides of the triangle.
3. Area of non-right triangle: $$\text{Area}=\frac{1}{2}ab\sin C$$

Worked Example

A triangular park has sides of length 50m and 70m, with an included angle of 50° between them. Calculate the length of the third side and the area of the park.

1. Use cosine rule for the third side: $$c^2=50^2+70^2-2(50)(70)\cos 50^{\circ }=2500+4900-7000(0.6428)\approx 2899.9$$ $$c\approx \sqrt{2899.9}\approx 53.9\text{ m}$$
2. Use area formula: $$\text{Area}=\frac{1}{2}(50)(70)\sin 50^{\circ }=1750(0.7660)\approx 1341{\text{ m}}^2$$

4. Three-figure bearings

Three-figure bearings are a standard way to measure direction in navigation, used frequently in IB AI SL context questions. A bearing is an angle measured clockwise from north, written as 3 digits with leading zeros if needed (e.g., 045° for northeast, 180° for south, 270° for west).

Key rules for bearings:

• When asked for the bearing of X from Y, the reference point is Y: stand at Y, face north, turn clockwise until you face X, that angle is the bearing.
• Bearings can be combined with sine/cosine rules to calculate distances between moving objects like planes, ships, or hikers.

Worked Example

A hiker walks 8km from their base camp on a bearing of 060°, then walks 10km on a bearing of 150°. Calculate the distance from the hiker to their base camp, and the bearing they need to follow to return directly to camp.

1. The angle between the two legs of the hike is $150^{\circ }-60^{\circ }=90^{\circ }$, so you can use Pythagoras for the return distance: $$d=\sqrt{8^2+10^2}=\sqrt{164}\approx 12.8\text{ km}$$
2. The angle between the first leg and the return path is $\arctan \left(\frac{10}{8}\right)\approx 51.3^{\circ }$. The return bearing is $60^{\circ }+180^{\circ }+51.3^{\circ }=291.3^{\circ }$, rounded to 291° as a three-figure bearing.

5. Surface area and volume

This subtopic covers 2D area, and 3D surface area and volume for standard shapes, plus scaling rules for similar figures. All formulas are given in your IB formula booklet, but you need to know when and how to apply them correctly.

Core rules:

• Prisms (including cylinders): Volume = area of cross section × length; surface area = sum of area of all faces
• Pyramids (including cones): Volume = $\frac{1}{3}$ × base area × perpendicular height
• Spheres: Volume = $\frac{4}{3}\pi r^3$; Surface area = $4\pi r^2$
• Similar figures: If the linear scale factor between two similar shapes is $k$, the area scale factor is $k^2$ and the volume scale factor is $k^3$. This is one of the most frequently tested rules in this subtopic.

Worked Example

Two similar cones have heights of 4cm and 12cm. The smaller cone has a volume of $30{\text{ cm}}^3$. Calculate the volume of the larger cone, and its total surface area if the smaller cone has a surface area of $45{\text{ cm}}^2$.

1. Linear scale factor $k=\frac{12}{4}=3$
2. Volume scale factor = $k^3=27$, so larger cone volume = $30\times 27=810{\text{ cm}}^3$
3. Area scale factor = $k^2=9$, so larger cone surface area = $45\times 9=405{\text{ cm}}^2$

6. Common Pitfalls (and how to avoid them)

• Pitfall 1: Using the sine rule when you have an included angle, leading to incorrect side/angle values. Why it happens: Students mix up use cases for sine and cosine rules. Correct move: If you have two sides and the angle between them, always use the cosine rule first.
• Pitfall 2: Forgetting to pad bearings to three digits, e.g., writing 45° instead of 045°. Why it happens: Lazy notation habits from pre-DP math. Correct move: Always add leading zeros to bearings to make them 3 digits, and double-check you are measuring clockwise from north.
• Pitfall 3: Calculating the largest empty circle in a Voronoi diagram as centered at a random point instead of a Voronoi vertex. Why it happens: Students forget that Voronoi vertices are the only points equidistant from three or more sites. Correct move: The center of the largest empty circle is always a Voronoi vertex, no exceptions.
• Pitfall 4: Using linear scale factor for volume/area calculations for similar shapes. Why it happens: Students skip checking the dimension of the measurement. Correct move: Circle if the question asks for length, area, or volume, then apply $k$, $k^2$, or $k^3$ respectively.
• Pitfall 5: Mixing up the reference point for bearings, e.g., calculating the bearing of A from B instead of B from A. Why it happens: Students skip highlighting the "from" term in the question. Correct move: Circle the reference point (the word after "from") before starting any bearing calculation.
• Pitfall 6: Forgetting to exclude open faces when calculating surface area, e.g., including the top of an open-top cylinder. Why it happens: Students use the standard surface area formula without reading the question context. Correct move: Tick off each face you are including in your surface area calculation to avoid overcounting.

7. Practice Questions (IB Math AI SL Style)

Question 1

A Voronoi diagram for four coffee shop sites has vertices at (2,3), (5,1), and (7,5) on a 1km grid. The distance from (5,1) to the nearest site is 1.8km, from (2,3) it is 2.1km, and from (7,5) it is 2.4km. What is the maximum radius of a new coffee shop that will not compete with existing sites?

Solution

The largest empty circle is centered at the Voronoi vertex with the largest distance to the nearest site, which is (7,5) with distance 2.4km. This is the maximum allowed radius for the new shop.

Question 2

A triangle has sides of length 6cm, 8cm, and 11cm. Calculate the size of the largest angle in the triangle, to the nearest degree.

Solution

The largest angle is opposite the longest side (11cm). Use the cosine rule: $$11^2=6^2+8^2-2(6)(8)\cos A$$ $$121=36+64-96\cos A$$ $$121=100-96\cos A$$ $$21=-96\cos A$$ $$\cos A=-\frac{21}{96}\approx -0.2188$$ $$A\approx \arccos (-0.2188)\approx 103^{\circ }$$

Question 3

A cylindrical water tank with an open top has a height of 2m and a diameter of 1.5m. Calculate the total volume of water it can hold, and the total external surface area that needs to be painted. Give answers to 2 decimal places.

Solution

1. Radius $r=0.75$m. Volume of cylinder = $\pi r^{2}h=\pi (0.75{)}^2(2)\approx 3.53{\text{ m}}^3$
2. Surface area = area of base + curved surface area = $\pi r^2+2\pi rh=\pi (0.75{)}^2+2\pi (0.75)(2)\approx 1.77+9.42=11.19{\text{ m}}^2$

8. Quick Reference Cheatsheet

9. What's Next

Geometry and Trigonometry is a foundational topic that connects to almost every other part of the IB AI SL syllabus. You will use Voronoi diagrams for spatial statistical modeling, trigonometric rules for periodic function modeling, and volume/surface area calculations for optimization and related rates problems in calculus. It is also one of the most common contexts for long-answer Paper 2 questions, which combine multiple topics to test your application skills.

If you have questions about specific problem types, exam grading conventions, or need more personalized practice to master these subtopics, you can reach out to Ollie on the homepage, where you will find custom quizzes, past paper breakdowns, and targeted revision plans aligned to your IB AI SL exam timeline.