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Rotating a Rectangle Around an Axis

Rotating a Rectangle Around an Axis
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2020-04-22 03:13
PolygonSolid GeometryTransformationGeometric ConstructionSlider Animation
① This worksheet demonstrates how rotating a rectangle about one of its edges (the axis) generates a three-dimensional solid, specifically a cylinder. ② Core mathematical concepts include solids of revolution, 3D geometry, the relationship between a rectangle and a cylinder, and rotational symmetry. ③ Construction details: GeoGebra commands used include Intersect, Point, OrthogonalLine, Polygon, Cylinder, Circle, and Rotate, with objects such as rectangle poly1 and its rotated copy poly1', along with auxiliary lines and points. ④ Suitable for middle and high school students learning introductory solid geometry or solids of revolution, and useful as a dynamic visual aid to understand how planar shapes generate 3D objects when rotated about an axis.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(yAxis) → B
  3. Point(xAxis) → C
  4. OrthogonalLine(B, yAxis) → a
  5. OrthogonalLine(C, xAxis) → b
  6. Intersect(a, b) → D
  7. Polygon(B, D, C, A) → poly1, b_1, d, c, a_1
  8. Cylinder(A, B, c) → e, f, g, h
  9. Circle(A, c, yAxis) → k
  10. Circle(B, c, yAxis) → p
  11. Rotate(B, (-r), yAxis) → B'
  12. Rotate(D, (-r), yAxis) → D'
  13. Rotate(C, (-r), yAxis) → C'
  14. Rotate(A, (-r), yAxis) → A'
  15. Polygon(B', D', C', A') → poly1', b', d', c', a'

AI Drawing Prompt Reference

In a 3D coordinate system, first construct a rectangle with its four vertices lying on the x-axis, y-axis, and the intersections of perpendicular lines drawn from those axes. Then, using one edge of the rectangle (the side on the y-axis) as the rotation axis, rotate the entire rectangle around this axis by an angle controlled by a slider, producing a rotated copy of the rectangle. Simultaneously, draw two circles of equal radius at the two ends of the rotation axis, and connect them to form a cylindrical surface, visually demonstrating the solid swept out by the rotating rectangle. The final figure should include the original rectangle, the rotated rectangle, the cylinder, and auxiliary axes and circles, highlighting how a planar shape generates a 3D solid through rotation.
The prompt above was compiled from the original drawing commands and is for reference only.