MathPlotAI Logo MathPlotAI

Cube Surface Area with Hemispheres - 3D Geometry Demo

Cube Surface Area with Hemispheres - 3D Geometry Demo
Loading the math board and drawing, please wait…
2017-12-28 18:48
Solid Geometry
This GeoGebra example demonstrates a 3D composite solid consisting of a cube with six hemispheres attached to its faces, focusing on the calculation of total surface area (aire). The core mathematical concepts involve solid geometry, surface area of cubes and spheres, and the additive property of areas for composite shapes. The construction uses the Surface command to generate curved surfaces, with objects including r (radius), c (cube), s1-s6 (hemispheres), f (a function or face), circles c1-c6, and dependent points. The model is built symmetrically to illustrate how the surface area of the combined shape relates to its components. This is suitable for middle or high school students learning about 3D geometry, surface area calculations, or for teachers demonstrating composite solids in class.

GeoGebra Commands & Parameters

  1. Surface(((r * cos(u)) * cos(v)), ((r * sin(u)) * cos(v)), c / 2 - (r * sin(v)), u, (-π), π, v, ((-π)) / 2, 0) → s1
  2. Surface(f((v * cos(u))), f((v * sin(u))), c / 2, u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c1
  3. Surface(f((v * cos(u))), f((v * sin(u))), ((-c)) / 2, u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c2
  4. Surface(c / 2, f((v * cos(u))), f((v * sin(u))), u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c3
  5. Surface(((-c)) / 2, f((v * cos(u))), f((v * sin(u))), u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c4
  6. Surface(f((v * cos(u))), c / 2, f((v * sin(u))), u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c5
  7. Surface(f((v * cos(u))), ((-c)) / 2, f((v * sin(u))), u, 0, (2 * π), v, abs(r), (c * sqrt(2) / 2)) → c6
  8. Surface(((r * cos(u)) * cos(v)), ((r * sin(u)) * cos(v)), ((-c)) / 2 + (r * sin(v)), u, (-π), π, v, ((-π)) / 2, 0) → s2
  9. Surface(c / 2 - (r * sin(v)), ((r * cos(u)) * cos(v)), ((r * sin(u)) * cos(v)), u, (-π), π, v, ((-π)) / 2, 0) → s3
  10. Surface(((-c)) / 2 + (r * sin(v)), ((r * cos(u)) * cos(v)), ((r * sin(u)) * cos(v)), u, (-π), π, v, ((-π)) / 2, 0) → s4
  11. Surface(((r * cos(u)) * cos(v)), c / 2 - (r * sin(v)), ((r * sin(u)) * cos(v)), u, (-π), π, v, ((-π)) / 2, 0) → s5
  12. Surface(((r * cos(u)) * cos(v)), ((-c)) / 2 + (r * sin(v)), ((r * sin(u)) * cos(v)), u, (-π), π, v, ((-π)) / 2, 0) → s6

AI Drawing Prompt Reference

Draw a 3D composite solid consisting of a cube with six hemispheres attached to each face, demonstrating the calculation of its total surface area. The cube and hemispheres are represented by smooth curved surfaces, and the entire structure is symmetric to illustrate how the surface area of the combined shape relates to its individual components.
The prompt above was compiled from the original drawing commands and is for reference only.