IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PlanesAngles Between Two Planes

Angles Between Two Planes (DP IB Analysis & Approaches (AA): HL): Revision Note

Written by: Amber

Reviewed by: Dan Finlay

Updated on 26 November 2025

Angle between two planes

How do I find the angle between two planes?

There are two pairs of equal angles formed when two planes intersect
The angles between two planes are equal to the angles between their normal vectors
The formula for an angle between two planes is $\cos θ = \frac{n_{1} \cdot n_{2}}{| n_{1} | | n_{2} |}$
- $n_{1}$ is a normal vector to one plane
- $n_{2}$ is a normal vector to the other plane
You can subtract one angle from 180° to find another angle

Diagram showing two intersecting planes with angle θ between them and their normal vectors. Labels explain the angle between planes and normal vectors. — **Example of angles between two planes**

Examiner Tips and Tricks

Always draw a sketch of the planes and their normals. This will make sure you do not confuse this method with the method for finding an angle between a line and a plane.

Worked Example

Find the acute angle between the two planes which can be defined by equations $Π_{1} : 2 x - y + 3 z = 7$ and $Π_{2} : x + 2 y - z = 20$ .

Answer:

3-11-3-ib-hl-aa-angle-between-two-planes-we-solution-2

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent