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Relationship between Base Length and Volume of a Square Pyramid (LR8-01-XT2)

Relationship between Base Length and Volume of a Square Pyramid (LR8-01-XT2)
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2019-05-22 20:13
FunctionSequenceSolid GeometryLimitGraphing
This interactive GeoGebra example demonstrates how the volume of a square-based pyramid changes as the base length varies, using a scatter plot and function graph. The core mathematical concept is the pyramid volume formula V = (1/3) × base area × height, showing a cubic relationship between base length and volume, with a limit as base length approaches zero. The construction uses GeoGebra commands: Polygon to create the square base, Pyramid to build the 3D solid, Sequence and Vector to generate base length values and corresponding volume data points, Zip to combine points into a scatter plot, Circle to mark the current point, RotateText for rotated labels, and Text for instructions. Suitable for middle school advanced or high school students exploring solid geometry, function graphing, and sequences.

GeoGebra Commands & Parameters

  1. LaTeX((((("\text{" * QUESTION) * "\\") * INSTRUCTIONS) * "}")) → TEXT
  2. Polygon({(0, 0), (0, 7), (11, 7), (11, 0)}) → viewBox
  3. Sequence(Segment[(i, -0.2), (i, 0.2)], i, 1, 10) → xTicks
  4. Sequence(i, i, xInc, 10, xInc) → xTickLabels
  5. Sequence(i, i, yInc, 180, yInc) → yTickLabels
  6. Polygon((0, 0), (0, baseLength), 4) → pyramidBase
  7. Vector((0, 0), (11, 0)) → u
  8. Vector((0, 0), (0, 7)) → v
  9. Sequence(Segment[(-0.2, i), (0.2, i)], i, 1, 6) → yTicks
  10. Sequence(Text[Element[xTickLabels, i], (i - 0.15, -0.7)], i, 1, 10) → xAxisTickLabels
  11. Sequence((i, volumePyramid(i)), i, 0, maxBaseLengthVisited, 0.5) → pointsToPlot
  12. Zip((x(A) / xInc, y(A) / yInc), A, pointsToPlot) → pointsToPlotAdjusted
  13. Circle(currentPoint, 0.3) → currentCircle
  14. RotateText(Text["\text{Volume}", (-1.75, 5)], 90°) → yAxisLabel
  15. Pyramid(pyramidBase, height) → pyramid, A, face1, face2, face3, face4, edge1, edge2, edge3, edge4
  16. Sequence(Text[Element[yTickLabels, i], (-1.1, i - 0.15)], i, 1, 6) → yAxisTickLabels
  17. Text("\text{Base Length}", (3, -1), false, true) → xAxisLabel

AI Drawing Prompt Reference

Draw a 2D coordinate system with the horizontal axis representing base length and the vertical axis representing volume, showing a curve starting from the origin that increases cubically. Place a 3D square-based pyramid diagram with adjustable base length, and overlay a scatter plot of volume values for different base lengths, highlighting the current data point with a circle. Label the axes as 'Base Length' and 'Volume', include tick marks and numbers, and add brief instructional text.
The prompt above was compiled from the original drawing commands and is for reference only.