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Mathematics
Angles and distances in space
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A 3D geometry lab: manipulate figures, construct auxiliary geometry and explain spatial angles and distances.
Drag empty space to rotate; use the camera controls to zoom and frame. Blue handles adjust the same values as the parameter sliders.
Preparing the 3D scene…
θ / Step 4 · 4
Where is the angle between SB and the base?
SA ⟂ (ABCD)AB = proj₍P₎(SB)θ = ∠SBA
Length SB
4 / 4
Identify the required angle
The required angle is ∠SBA. For a perpendicular line, the projection is one point and the angle is 90°.
Geometric explanation
A is the perpendicular projection of S; B already lies in the base plane. Thus AB is the projection of SB and the required angle is ∠SBA. Changing the view does not change the angle.
Common misconception
The angle between SB and SA uses the normal to the base, not the projection of SB.
Try and explain
Drag S upwards while keeping AB = 4. Does the angle increase? Try SA = 3 for a 3–4–5 triangle.
Model notes
- Lines and planes are infinite; visible segments and patches are representations. A perpendicular foot may lie on an extension.
- Angles between lines, a line and a plane, or two planes range from 0° to 90°. A dihedral depends on the selected half-planes and may be obtuse.
- Lengths use arbitrary units. Readings are rounded to one decimal place; calculations retain greater precision.
- Handles change constrained parameters to preserve required geometric relationships. Degenerate configurations are explained separately.
- The lab uses curated configurations. Numerical observations do not replace a mathematical proof.