Here's a fun demonstration from Cornell maths professor Steven Strogatz. Take a clementine (or any spherical, peelable fruit) and trace around its widest part four times. Then peel it. Flatten out the peelings as best you can and divvy them up evenly among the circles. Voilà! Tangible proof that the the surface area of a sphere is 4πr2!

Photo Credit: arbyreed | CC BY-NC-SA 2.0

Strogatz calls it "Proof by Clementine":

Such a simple demonstration – but so effective!

If you're feeling extra teacherly, you can use the peelings from your fruit to segue into a lesson on Guassian curvature: When you're divvying up your orange peels, you may notice that it's impossible to flatten them out without stretching or tearing them. That's because spheres (like the intact peel of an orange) and flat surfaces have different Gaussian curvatures. This is the same reason a 2-D map can never perfectly preserve the relative size and shape of Earth's assorted land masses.

If you're into this kind of thing, I highly recommend following Strogatz on twitter. He posts this kind of stuff pretty regularly.