www.thinkmathematics.com Great ideas and resources for teaching engaging mathematics lessons

Great ideas and resources for teaching engaging mathematics lessons

Torricelli's Trumpet


This activity explores a famous paradox -Torricelli's trumpet or Gabriel's horn. It is taken from rotating the graph of y= \frac{1} {x}, x \ge 1 about the x axis. It is sometimes called the painter's paradox and hopefully you will see why. Evangelista Torricelli lived in a time before calculus but considered this using the method of indivisibles . You should be familiar with integration and be able to find volumes by rotating areas about the x axes.

Teachers Notes - Why? How? What?

Why we like this activity....
Students get to prove previously assumed formulae for volume and surface area of spheres. This is a great paradox -
it takes an infinite amount of paint to paint an infinite area, but the horn can be filled with a finite amount of paint.

How this activity can be used....
Provided that students have studied volumes of revolution this can be set as a homework or classroom activity.

What to expect when using this activity, from our experience...
Lots of discussion! Look at "Visualising Volumes of Revolution" activity.

Extra notes


Author/Date Richard Wade 04.10.09

Credits Evangelista Torricelli


mathematics 16-18 calculus integration volume revolution paradox