Knowledge organisers

Geometry and trigonometry: IB Maths AI HL 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

Graph: A set of vertices joined by edges; the degree of a vertex is the number of edge ends meeting it.
Eulerian circuit: A closed walk using every edge exactly once; a connected graph has one when every degree is even.
Minimum spanning tree: The cheapest set of edges that joins every vertex with no cycles (Kruskal's or Prim's algorithm).
Adjacency matrix: A table whose entry in row i, column j is the number of edges joining vertex i to vertex j.

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; cone; sphereIn the formula booklet	\(V=\tfrac13Ah;\) \(V=\tfrac13\pi r^2h,\) \(A=\pi rl;\) \(V=\tfrac43\pi r^3,\) \(A=4\pi r^2\)
Sine, cosine rules; areaIn the formula booklet	\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\) \(c^2=a^2+b^2-2ab\cos C,\) \(A=\tfrac12ab\sin C\)
Arc; sector, degreesIn the formula booklet	\(l=\tfrac{\theta}{360}2\pi r,\) \(A=\tfrac{\theta}{360}\pi r^2\)
Arc; sector, radiansIn the formula booklet	\(l=r\theta,\) \(A=\tfrac12r^2\theta\)
IdentitiesIn the formula booklet	\(\cos^2\theta+\sin^2\theta=1,\) \(\tan\theta=\frac{\sin\theta}{\cos\theta}\)
Radians; bearings	\(\pi\text{ rad}=180^\circ;\) \(\text{bearings clockwise from north}\)
Perpendicular bisectors & Voronoi: cell edges on perpendicular bisectors of sites; largest empty circle at a vertex or on the boundary
Reflection in \(y=(\tan\theta)x\)In the formula booklet	\(\begin{pmatrix}\cos2\theta&\sin2\theta\\\sin2\theta&-\cos2\theta\end{pmatrix}\)
Stretch ×\(k\): horizontal, vertical; enlargementIn the formula booklet	\(\begin{pmatrix}k&0\\0&1\end{pmatrix},\) \(\begin{pmatrix}1&0\\0&k\end{pmatrix};\) \(\begin{pmatrix}k&0\\0&k\end{pmatrix}\)
Rotation about \(O\) by \(\theta\): anticlockwise; clockwiseIn the formula booklet	\(\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix};\) \(\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}\)
Area of image	\(\|\det A\|\times\text{area of object}\)
Composite transformation: \(B\) then \(A\) is \(AB\)
Vector magnitudeIn the formula booklet	\(\|\boldsymbol v\|=\sqrt{v_1^2+v_2^2+v_3^2}\)
Scalar product; angleIn the formula booklet	\(\boldsymbol v\cdot\boldsymbol w=v_1w_1+v_2w_2+v_3w_3=\|\boldsymbol v\|\|\boldsymbol w\|\cos\theta\)

More formulas are on the full IB Maths AI HL formula sheet.

Worked example

A connected graph has vertex degrees 2, 4, 3, 3 and 2. How many edges does it have, and does it have an Eulerian circuit?

The degrees add to twice the number of edges: 2 + 4 + 3 + 3 + 2 = 14
Two vertices have odd degree

Answer: 7 edges. No Eulerian circuit, but there is an Eulerian trail between the two odd vertices

Common mistakes

IB notation: AB as a length, vectors in columns, f⁻¹ vs f′, the variables asked for
Graph algorithms: Prim vs nearest neighbour, no method shown, bounds mixed up
Using the 2D distance formula without the extra z-term in 3D
Using the whole circle (not the fraction θ/360) when finding an arc length or sector area

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.
Use the cosine rule to find a side from two sides and the included angle, and an angle from three sides.
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.
Use the language of graph theory, find the degree of each vertex and recognise simple, complete and connected graphs.
Use the nearest-neighbour algorithm to find an upper bound for the travelling salesman problem, starting from different vertices to improve it.
Build a table of least distances and use the nearest-neighbour and deleted-vertex algorithms to bound the best tour.

The printable sheet

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

Revise it next

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