In this hands-on Explorer lesson students learn the derivation of the surface area formula for a sphere.
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Author: Laura Jewell
Here is a concrete way to show students why the surface area of a sphere is derived from the formula 4 pi x r2.
Students should already understand that the surface area of an object can be represented by how much wrapping paper it would take to cover it. Ask them to then picture a sphere (a balloon or ball) and a piece of paper that is cut as wide as its diameter and as long as its circumference.
If you wrap the ball with the paper, you see that it would cover the entire sphere if it weren't for all the overlaps (which would fit into the gaps if you cut them out).
The formula for the surface area of the paper is:
length x width = circumference x diameterThis is easily understood by looking at the picture above. Now substitute formulas we know:
C = 2 pi x r and d = 2 r
C x d = 4 pi x r2 = surface area
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