Sections in the sphere
In the initial position of the applet it represents a circle, and when you move the cursor changes the vertical position of a chord. You could here think about the arithmetic mean and the geometric mean of two magnitudes and see that the geometric mean is less than or equal to the arithmetic mean. Both agree only if the two quantities are equal.
But here we see the sections in one sphere. Simply click on the picture and drag to see the representation of a sphere and one section.
If x is the distance from the center of the sphere to the section, the radius of the section is
And the area of the section is
The vertical cursor allows us to change the height of the section.
If we click and drag on the figure we can rotate it.
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The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
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Sections on a tetrahedron
Special sections of a tetrahedron are rectangles (and even squares)
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Sections in Howard Eves's tetrahedron
Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence
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Surprising Cavalieri congruence between a sphere and a tetrahedronn
We show a sphere and the Howard Eves's tetrahedron with congruent sections.
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