Impossible Triangles

Can these triangles exist?

The main thrust of this activity is easily explained - What reasoning can you use to decide if these triangles can exist or not? [1]This is an alternative way to apply what you know about triangle geometry. Pythagoras's theorem, SOHCAHTOA, Sine rule, Cosine rule and more can all be used to reason one way or another and the whole point is for students to enter in to discussion with each other! There are lots of ways to approach this task most of which revolve around the notion of 'IF this is true, THEN this must be true...


This activity is all about reasoning and using knowledge about triangle geometry to do so. In doing so, students should get lots of practice of applying Pythagoras and Trigonometry.


The Impossible Triangles can be printed of this activity sheet and screen shots are included below. Teachers can see the solutions and read a bit about the activity on the Impossible Triangles teacher notes.

Syllabus links

New - Sections, Presumed knowledge, 5.2, 5.3 Previous - Sections 5.3 and 5.4


  • I think this activity works best without any introduction from the teacher or clues about how to go about it. Simply hand out the worksheet and ask students to say which of the triangles can exist.
  • Alternatively, teachers might choose to start with an example as a whole class and discuss
  • Once finished, you might choose to split the class in two and have a little competition, where teams take turns to pick a letter. They score a point if they have correctly judged it possible or not!

I did it my way!

As a practising maths teacher I know that most of us like to give activities our own little twist and do them 'our way'. It would be great to add a little collection of 'twists' from users. You can either add your twist to the comments section below where everyone can see them or add them by clicking the 'submit feedback' button above.


  • 1. To do the activity you need to assume that all side lengths are rounded to the nearest whole number!
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