Category Archives: trigonometry

Concentric Equilateral Triangles

The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.

Screen Shot 2017-04-14 at 23.57.47

What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm.  What is the ratio of heights now?

Screen Shot 2017-04-15 at 00.06.54

Explore what happens for other border widths.  Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.




The angle defines the ratio of side lengths in a right angle triangle

This post shows how to use Geogebra to demonstrate this fundamental truth in geometry and hopefully demystify Trigonometry to a certain extent.

As with all things Geogebra, I always try to start with a blank sheet (see other posts on this here and here).  This time, I’m not using the Geogebra app itself but just launching it from within a Chrome browser window which works pretty well.


Once it is launched, I right click in the middle to remove the axes, but I am going to leave the grid on.

Then I create the triangle by constructing a line, a perpendicular line…


…and a third point which I then join to create a triangle using the polygon tool.


Next, measure the base angle of the triangle remembering the convention that angles are measured in an anti-clockwise direction.


The next bit is a tad fiddly. You need to right click on the line segment to change the label to “value”.  Then do the same for the other two sides of the triangle so that you now have one angle and all three side labelled.


So far, this has taken about 2 minutes to create from a blank screen.  You could do it in advance of the lesson, but I think it is worth doing it in front of the class, maybe having practiced it a couple of times.  Using “something I created earlier” is less powerful – it looks like some sort of trick, somehow.

Now you have everything set up you can start moving the points as shown here.


I start by moving point B, thus keeping the angle fixed.  I would ensure students have calculators in front of them and ask them to calculate opposite divided by adjacent. Then move the triangle to get different values for side lengths. Then do the calculation again. The answer is the same, of course.  I might ask them how they could get that directly from the angle (tan angle).  Depending on where the discussion goes with that, I might then move on to look at sin and cos.

Finally, I always like to talk about how things were done in the old days, being careful to point out that I’m not that old and that I didn’t actually use these…crc_trig_tables.jpg

I explain that the sin button on your calculator is basically just looking up the values in the sin column of a table like this – not actually true, I know, but it helps understand what’s going on so that’s OK for me!



The paper napkin trigonometry trick with a smattering of Pythagorean triples. 

Take a piece of paper and do the following:

  1. Make it into a square (interesting discussion on best way to do this).
  2. Fold in half then unfold so you have created crease along a vertical line of symmetry
  3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
  4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??


Pythagoras and Trigonometry Revision

I used these cards for revision of Pythag and Trig recently.


They are really nice and there are lots of them so you can decide how far you want to go. I just have the file and I’m not sure who created them, so if you know, let me know so I can credit them / link directly.

This sort of activity works really well with a class that will have productive and supportive conversations about the maths and enjoy challenging each other.  It gets them out of their seats and they start to get a sense of which questions are straightforward and which will present more of a challenge. I’m on the look out for similar things for other topics.

I’ll be setting this for homework as it has explanations as well as examples all in one place:Capture