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GeoGebra几何作图：立方体与球体体积比较

GeoGebra几何作图：立方体与球体体积比较
几何画板及绘图内容加载中，请稍等……
2015-06-16 04:13
立体几何几何作图滑块动画
本范例演示如何通过GeoGebra软件构造立方体和球体，并比较它们的体积。核心数学概念包括立体几何中的体积计算、几何变换以及参数化建模。作图要点包括使用Corner、Cube、Sphere等命令创建几何对象，并通过TableText和RandomizationText进行数据可视化。适合高中数学课程或几何学习者，用于理解三维几何与体积关系。

GeoGebra 命令及参数

  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 绘图提示语参考

绘制一个立方体和一个球体，立方体的边长分别为小立方体和大立方体，小立方体边长为 2 厘米，大立方体边长为 4 厘米。球体的半径根据立方体的体积计算得出，小立方体对应的球体半径约为 1.13 厘米，大立方体对应的球体半径约为 1.89 厘米。确保小立方体位于左下角，大立方体位于右上角，球体分别与对应的立方体等体积。
以上提示语为根据原始绘图命令整理而来，仅供参考。