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GeoGebra Geometric Construction: Cube and Sphere Volume Comparison

GeoGebra Geometric Construction: Cube and Sphere Volume Comparison
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2015-06-16 04:13
Solid GeometryGeometric ConstructionSlider Animation
This example demonstrates constructing cubes and spheres using GeoGebra and comparing their volumes. Key mathematical concepts include volume calculation in solid geometry, geometric transformations, and parametric modeling. Construction involves commands like Corner, Cube, Sphere, and TableText for visualization. Suitable for high school mathematics courses or geometry learners to understand three-dimensional geometry and volume relationships.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  3. Corner(1) → A
  4. Corner(3) → B
  5. Cube(((-hs), (-hs), (-hs)), (hs, (-hs), (-hs)), xOyPlane) → b, C, face4, faceCDHG, faceEFGH, edge2, edge1, edge4, edgeCD, edge3, edgeDH, edgeEH, D, edgeEF, edge5, edgeFG, edgeCG, edgeGH, E, F, G, H, face1, face3, face2
  6. Sphere((0, 0, 0), 0.62) → a
  7. LaTeX((((("\text{small cubes: side length of }\textcolor{#000099}{" * s_s) * "}\text{ cm}\cr\text{large cubes: side length of }\textcolor{#000099}{") * s_l) * "}\text{ cm}")) → Problem
  8. LaTeX("\text{Move the slider to solve the problem.}") → text0
  9. LaTeX("\text{Find a formula the company can use to determine the radius of a sugar sphere with}\cr\text{the same volume as a sugar cube of side length }\textcolor{#000099}{\mathit{x}}.") → text1
  10. LaTeX("\text{Recall the formula for the volume, }\textcolor{#000099}{\mathit{V}_{sphere}}\text{, of a sphere of radius }\textcolor{#000099}{\mathit{r}}.") → text2
  11. LaTeX("\textcolor{#000099}{\mathit{V}_{sphere}=\frac{4}{3}\pi \mathit{r}^3}") → text3
  12. LaTeX("\text{Recall the formula for the volume, }\textcolor{#000099}{\mathit{V}_{cube}}\text{, of a cube of side length }\textcolor{#000099}{\mathit{x}}.") → text4
  13. LaTeX("\textcolor{#000099}{\mathit{V}_{cube}= \mathit{x}^3}") → text5
  14. LaTeX("\text{To find the radius of a sphere with the same volume as a cube of side length }\textcolor{#000099}{\mathit{x}},\cr\text{set the volume formulas equal, and solve for the radius }\textcolor{#000099}{\mathit{r}}.") → text6
  15. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}}) → text7
  16. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}}) → text8
  17. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}}) → text9
  18. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}}) → text10
  19. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}}) → text11
  20. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}}\sqrt[3]{\mathit{x}^3}}"], LaTeX["\text{Simplify the right side into a product of cube roots.}"]}}) → text12
  21. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}}\sqrt[3]{\mathit{x}^3}}"], LaTeX["\text{Simplify the right side into a product of cube roots.}"]}, {LaTeX["\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right) \mathit{x}}"], LaTeX["\text{Simplify.}"]}}) → text13
  22. LaTeX("\text{Thus, a sphere with radius }\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\mathit{x}}\text{ has the same volume as a cube with side length }\textcolor{#000099}{\mathit{x}}.") → text14
  23. LaTeX("\text{Use the formula }\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\mathit{x}}\text{ to determine the radii required for the proposed sugar spheres.}") → text15
  24. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[""]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[""]}}, "c|_") → text16
  25. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[(("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_s) * "\right)}")]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[(("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_l) * "\right)}")]}}, "c|_") → text17
  26. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[(((("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_s) * "\right)\approx") * r_s) * "}")]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[(((("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_l) * "\right)\approx") * r_l) * "}")]}}, "c|_") → text18

AI Drawing Prompt Reference

Draw a cube and a sphere, with the cube's side lengths being for small and large cubes, 2 cm and 4 cm respectively. The radius of the sphere is calculated based on the volume of the cube, approximately 1.13 cm for the small cube and 1.89 cm for the large cube. Place the small cube in the lower left corner and the large cube in the upper right corner, with the spheres having the same volume as their respective cubes.
The prompt above was compiled from the original drawing commands and is for reference only.