MATHEMATICS LAB · GEOMETRY

Study of Geometric Solids

Explore oblique prisms and 3D solids; animate base displacement and study cross-sections, surface areas and volumes.

Measurements use arbitrary units (u). The horizontal plane cuts the solid at the chosen height. For a sphere the height is measured from the bottom pole; a sphere has no base.

1. Define the solid

Type and dimensions

Geometric solid

Side s (u) Length a (u) Width b (u) Number of base sides n Radius r (u) Height h (u)

Upper base shift Δx (u) Upper base shift Δy (u)

A cube has six congruent square faces and twelve equal edges.

Cutting plane

Cutting-plane height: 50%

Solids and horizontal sections Oblique prisms and Cavalieri's principle Inclined-plane sections

2. Interactive three-dimensional construction

Rotatable view · movable cutting plane

CubeDrag to rotate the scene or use the left-hand slider to change the section height.

Drag to rotate the solid. Use the mouse wheel or + and − buttons to zoom. Arrow keys rotate the view. The height slider and inclination controls move the cutting plane and update the section and its area.

Surface Section of the finite solid Height Hidden edges

3. How to read the construction

4. Results

Live update

5. Explained calculations

Formulas and numerical substitutions

What you can explore

Polyhedra

Cubes, cuboids, right and oblique prisms, and pyramids: edges, heights, cross-sections and Cavalieri's principle.

Solids of revolution

Cylinder, cone and sphere: radii, slant heights, surface areas and volumes.

Sections of solids

Horizontal and inclined cuts. Ideal double cone, conic sections and the area of finite-solid cross-sections.

Curved surfaces and oblique-section contours are approximated graphically. For polyhedra, intersections are calculated from face edges; oblique sections of cones and cylinders use numerical area approximations. The additional double cone is an ideal educational construction, not part of the finite solid. Nets of solids are available in the dedicated tool.