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Using Cross Section Flyer, I was able to find the radius and diameter at different points of the cone, Use this, I knew that a cone was similar to a cylinder, so I used the formula for finding the volume of a cylinder as my base. I took the different radii as my cylinders and averaged to find the volume of my irregular shape.

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The post above this clarifies what I meant for the post next to this. For the post directly beneath mine, I don't have an actual estimate here since I misread the directions and thought they meant to account for the changes in the base / radius / height.

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I think Benito’s idea is really good. When teaching students the measurement of cone volume, the question—how does a cone form? –should be emphasized. But sorry I don’t quite understand how Benito gets to 5(2r)h. Could you explain more about it?

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Can you think of a way to explain your process geometrically that would not involve Calculus?

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I’m thinking about the volume of a cone is made up of numerous different 2-D circles.

As the picture shows, I make the slice shape a circle and find its area as r² (r ranges from 0 to the radius (R) of the bottom circle), then the volume of the cone is (02 + 1² + 22 +……..R²) (here we only use integer radius to estimate the cone volume).

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This is an interesting method, it will be interesting to see how your estimated volume compares to task 2

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Maybe thinking about horizontal cross sections instead of vertical might be helpful?

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The more cylinders you have, the more accurate the approximation is. I think 4 would be fine for a rough estimate. you would have to exclude some volume near the point and work your way from there

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Graham and Jennifer have a great way of having a conversation: See "Graham" to "Jennifer" Title. 

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Graham and Jennifer have a great way of having a conversation: See "Graham" to "Jennifer" Title. 

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How would you decide to partition your cylinder into 2 or 4 cross sections? Where would you make those Partitions?

Is there a partition into 2 or 4 that might be more accurate than another?

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Seems like what I did too.

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I played with Cross Section Flyer and I tried to come up with a general solution different from what I had in my mind (cone is one third of a cylinder)using portioning the cone into many circles which are close to each other and then use the formula for cylinder to estimate the volume of the cone. I think based on the numbers of circles (or small cylinders inside the cone) the estimation can be more or less accurate. With more cylinders we have less free space between cone and cylinders.

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We can divide the cone into cylinders, make a choice regarding the radius of the cylinder (where the top or the bottom of the cylinder passes through the cone, an average of the top or bottom radii, or a weighted average), calculate the volume of each cylinder, then sum the cylinder volumes to estimate the cone volume.

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Finding the volume of a cone by cross sections looked difficult to me at first.  Then I starting thinking about calculus and how you might be able to use some calculus in trying to estimate the volume.  If you partition the cone into 2, 4 or more pieces, you could integrate the pieces.  Which essentially is what integration is as it takes small pieces of area and adds them together.  I would try to make horizontal cross sections through the cone, with a known height and radius.  Integrate each part and add up the pieces to get an estimation.  I’m sure there are other ways, but that is what came to mind first.

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At first, I was thinking about finding the height of the cone and using what I know about area of triangles to help find the volume. I was then thinking of the volume of a pyramid, lwh/3, but using the diameter of the circle instead...then I lost my way. I'm not sure what to do now...

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I was thinking that if we had this "triangle" type cross section, we could rotate it by 2π that we would have created the cone. So the base of this triangle is 2r, so the area would be .5(2r)h=rh, then we rotate so the volume is 2πrh.

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We can take cylindrical cross sections and add those volumes to get the volume of the cone.

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What are some of the methods/results from your investigations? Conjecture about the accuracy of your estimation given the number of partitions.

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What are some of the methods/results from your investigations? Conjecture about the accuracy of your estimation given the number of partitions.

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How would you develop a similar method to find the volume of a cone?

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How would you develop a similar method to find the volume of a cone?