Sunday, November 13, 2011

Volume of a cylinder & The sum of Areas

“Why do we need to know this?” How often do we get these questions in the classroom? Sometimes in order for students to see the value in mathematics you must take it away from them.




Take for example the formula for the volume of a cylinder, V = (pi)(r^2)(h)

If you take away the (h) within the equation you are left with the equation for the area of a circle. Thus, as we know, a cylinder is simply a bunch of circles stacked over top of one another. But, do your students know this?




Here’s an exercise,

1) Bring in a printable worksheet with numerous circles of equal size
2) As your students to cut out each circle

3) Next, ask them to find the sum of all the areas of each circle they cut out

4) Next, ask them to stack all the circles and use a ruler to measure its height

5) Finally, ask them to find the volume of the cylinder using V = (pi)(r^2)(h)

6) Does it closely resemble the sum of areas they previously calculated?

7) Wasn’t it a lot faster?



3 comments:

Lsquared said...

This doesn't work unless each circle is 1 unit high, which, if you print them on paper, is pretty unlikely.

Jeremiah Dyke said...

Certainly there would be a margin of error here. My thought behind the idea was to make the connection between the two formulas.

Manjit said...

I think the author of this post is right. The given argument or relation always works because the circle is a planner shape and there is no need to think about its height (there is no height for two dimensional shapes). LSquared took it a little wrong, as this concept always work ( Volume of solids = area of the base x the height of solid).

The is no concept of height (third dimension) in planner geometry, hence no question of unit height of a circle.

The formula to find the volume of a cylinder is driven from area of base circle multiplying to its height, the same way as the formula to find the volume of any cube or cuboid is derived by multiplying the area of the bottom face to the height of the solid.

Manjit
area of the circle

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