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Volume of Sphere by Lttcmooc

Volume of Sphere by Lttcmooc
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2016-10-07 15:35
Solid GeometryCalculusGeometric Construction
This example demonstrates the calculation of a sphere's volume. Key mathematical concepts include the volume formula of a sphere. Using GeoGebra commands such as Sphere, Sequence, and Intersect, the construction process of the sphere is dynamically shown. Suitable for high school mathematics, particularly in solid geometry and calculus courses.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), r) → a
  2. Sequence(z = (-r) + ((2 * r) * i / n), i, 1, n - 1) → list1
  3. Sequence(IntersectPath[Element[list1, i], a], i, 1, n - 1) → list2
  4. Sequence(Circle[(0, 0, r), (0, 0, (-r)), ((r * cos(((2 * π) * i / n))), (r * sin(((2 * π) * i / n))), 0)], i, 0, n - 1) → list3
  5. Intersect(Element[list2, floor(n / 2)], Element[list3, 1]) → a1_1, a1_2, ,
  6. Intersect(Element[list2, floor(n / 2) + 1], Element[list3, 1]) → a2_1, a2_2, ,
  7. Intersect(Element[list2, floor(n / 2)], Element[list3, 2]) → a3_1, a3_2, ,
  8. Intersect(Element[list2, floor(n / 2) + 1], Element[list3, 2]) → a4_1, a4_2, ,
  9. Polygon(a2_1, a1_1, a3_1, a4_1) → poly1, a2_3, a1_3, a3_3, a4_3
  10. Polygon(a2_1, A, a4_1) → poly2, a4_4, a2_4, a_1
  11. Polygon(a4_1, a3_1, A) → poly3, a_2, a4_5, a3_4
  12. Polygon(A, a1_1, a2_1) → poly4, a2_5, a_3, a1_4

AI Drawing Prompt Reference

Draw a sphere of radius r centered at the origin. Inside the sphere, create a series of horizontal planes equally spaced from bottom to top, intersecting the sphere to form latitude circles. Also, draw several great circles evenly distributed around the vertical axis, intersecting the latitude circles to create a grid of points on the sphere. Select the region bounded by two adjacent latitude circles and two adjacent great circles, and connect its four intersection points to form a spatial quadrilateral patch. Then, using one edge of this patch and the sphere's center, construct two triangular patches covering the adjacent spherical surface areas, illustrating the volume element decomposition.
The prompt above was compiled from the original drawing commands and is for reference only.