Surface area of spheres- an apeeling lesson…

How do you give pupils a beautiful visual picture of why the surface area of a sphere is 4 pi r squared? A super lesson idea I heard recently involved an orange!

Give pupils an orange and a blank sheet of paper. They have to draw circles around the orange onto the paper. The idea is that the circles on the paper should have the same radius as the orange. They draw as many circles on the paper as possible.

Then get the pupils to peel the orange and arrange the peel so that it fills the circles they drew on the paper. Some careful ‘sculpting’ of the peel will be required to get it to fill the circles with no gaps or overlaps. Pupils should fully fill one circle with peel before moving onto the next circle and so on.

A beautiful image should arise whereby the peel completely fills four of the circles proving the surface area of the sphere is 4 pi r squared!

If you give this a go with your class please take some pictures and email them to us at so we can feature them on this post!

Finally, time for an orange joke:

What did the little chic say when it’s mother laid an orange?

Look at the orange marmalade šŸ™‚

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6 Responses

  1. Dani Quinn says:

    I love this! Any thoughts on volume of spheres?

    • Don Cohen says:

      Volume Relationship Set – Ideal School Supply Co., Alsip, IL 60482

      The set contains other shapes, but we use the cylinder and the sphere. We assume the volume of the cylinder is = Pi*r^2*h. The sphere has the radius = to the radius of the cylinder, and the height of the cylinder = 2*r of the sphere. You fill the cylinder with water that’s in a bucket. Then stuff the sphere into the cylinder. Using Achemedes’ law, the amount of water displaced = the volume of the sphere. You look at the volume of water left in the cylinder and guess about 1/3 full. That means the vol. of the sphere is 2/3 the Vol of the cylinder.. So the vol.of the Sphere = (2/3)*Pi*R^2*2*r = (4/3)*Pi*r^3 which is correct.

      One can find the Vol of a cone compared to the vol of a cylinder, and a pyramid compared to a cube. The pieces are sturdy and have lasted for quite a while.

  2. Looks like an excellent lesson. šŸ™‚

  3. Marshall says:

    Not sure about the lesson but that joke is pure gold!

    OK, the lesson is pretty good too.

  4. Lloyd Knox says:

    Just found your post after we did a similar exercise at home that was also kind of opposite. Opposite because we tried to cover a curved thing with flat things. Instead of oranges and a sheet of paper we used pancakes and a basketball:

  1. 02/12/2016

    […] about the surface area of a sphere using an orange that you unpeel. It’s described on this wonderful blog by William Emeny. (His blog’s name is eerily similar to my […]

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