Knowledge organisers

Geometry and trigonometry: IB Maths AI SL knowledge organiser

Everything to know about geometry and trigonometry on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Bearing: A direction measured clockwise from north, written with three figures, such as 045°.
Angle of elevation: The angle measured up from the horizontal to the line of sight.
Perpendicular bisector: The line at right angles to a segment through its midpoint: every point on it is the same distance from both ends.
Voronoi diagram: Splits a plane into cells, each holding the points nearer to one site than to any other.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

3D distance (\(\Delta x=x_1-x_2\), …)In the formula booklet	\(d=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}\)
3D midpointIn the formula booklet	\(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2,\tfrac{z_1+z_2}2\right)\)
Pyramid; coneIn the formula booklet	\(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(\text{curved }A=\pi rl\)
SphereIn the formula booklet	\(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Slant height of a cone	\(l=\sqrt{r^2+h^2}\)
Sine ruleIn the formula booklet	\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
Cosine ruleIn the formula booklet	\(c^2=a^2+b^2-2ab\cos C,\) \(\cos C=\frac{a^2+b^2-c^2}{2ab}\)
Area of a triangleIn the formula booklet	\(A=\tfrac12ab\sin C\)
Arc; sector (\(\theta\) in degrees)In the formula booklet	\(l=\frac{\theta}{360}\times2\pi r,\) \(A=\frac{\theta}{360}\times\pi r^2\)
Elevation / depression measured from the horizontal; bearings clockwise from north, 3 figures
Perpendicular bisector of \(AB\): through the midpoint of \(AB\), gradient \(-\tfrac1{m_{AB}}\)
Voronoi: cell edges lie on perpendicular bisectors of pairs of sites; the point furthest from every site ("toxic waste dump") is at a vertex or on the boundary

Worked example

In triangle ABC, a = 7 cm, b = 8 cm and angle C = 60°. Find c.

Cosine rule: c² = a² + b² − 2ab cos C
c² = 49 + 64 − 2 × 7 × 8 × 0.5 = 57

Answer: c = √57 = 7.55 cm (3 s.f.)

Common mistakes

Voronoi diagrams and bisectors: midpoint, negative reciprocal, vertices
Triangles and 3D: wrong rule, wrong angle, missing units
Using the wrong right-angled triangle when isolating a 3D angle
Forgetting to check whether the GDC is in degree or radian mode

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the distance between two points and the midpoint of a line segment in three dimensions, using the 3D form of Pythagoras' theorem.
Find the angle between two lines and between a line and a plane in 3D solids by identifying and solving the right-angled triangles inside them.
Use the sine rule to find missing sides and angles in non-right-angled triangles, in degrees, drawing and labelling a diagram for each problem.
Find the equation of the perpendicular bisector of two points and use bisectors to construct the cells of a Voronoi diagram for a few sites.
Update a Voronoi diagram when a new site is added, and solve the toxic waste dump problem by finding the point furthest from every site.
Solve multi-stage navigation problems with three-figure bearings, using the sine and cosine rules to find distances and directions.

The printable sheet

Geometry and trigonometry knowledge organiser for IB Maths AI SL: one A4 page of key definitions, formulas, a worked example and common mistakes — Geometry and trigonometry knowledge organiser (IB Maths AI SL), A4. Download the PDF.

Revise it next

Other IB Maths AI SL topics: Number and algebra · Functions · Statistics and probability · Calculus · All IB Maths AI SL organisers