I have already worked both out, but what I don't understand is why they (triangle and cone) have differing heights and bases.Surely they should be the same.The answers I have are: 1.5 and 3^0.5/2 for the triangle and circle.4/3 and (2*2^0.5)/3 for the cone and sphere.When the measurements of the cone are converted into the 2D plane, the triangle formed is smaller. Yet when both measurements are converted into cones, it is bigger.When I visualise it, it doesn't make sense how a triangle with a larger area spun round has a smaller volume.Could someone explain it in way that makes sense logically.Thanks.
Biggest triangle in circle of radius one is different as a cone and sphere
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lev888
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My guess: cover the triangle by a bunch of small 'tiles' (e.g. squares and triangles). Image the 3d shape obtained by spinning each tile. You'll get donuts with square or triangular cross-sections. Which of them produce large donuts? The ones farther from the axis of revolution. How do you maximize the total volume? More tiles farther from the axis. This makes the cone lower and fatter.
Dr.Peterson
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A nice way of looking at it!
The essential idea here is that the volume of the cone is not proportional to the area of the triangle, because different parts of the triangle contribute differently to the volume; so you should not expect the volume to be maximized when the area is.
For some related insights, look up Pappus' Centroid Theorem. This implies that, to maximize the volume, you will want to get more of the triangle farther from the axis, just as lev888 said (though doing do also decreases the area, so it is not just a matter of making the triangle as wide as possible).
the simple answer is that the radius of the cone is squared, and multiplied by pi.
or/ compared with the base of a triangle, the base of the cone is a disc.
and you sacrifice some ht for this added area to give you maximum volume.
my answer for the triangle ht is the same as yours (1.5), but I did not get the same answer for the cone. My answer for cone ht=1.25, cone radius=(15)1/2/4. [My (max) triangle's base is 31/2 which is what i think you meant, you gave 1/2 of that.]
similarly though, we agreed the cone ht is shorter than triangle, and its base is wider.
If you think about it this makes sense, as the goal is to maximize volume of the cone, i.e. the circular discs (added together, bottom to top) have larger diameter.
intuitively it may make sense that if one finds a maximum triangle area inside a circle and spin that around it will be the maximum cone volume within a sphere, but as we see, that ain't so.