Angle Between a Line and a Plane, Distance in a Regular Pyramid
Mathematics3D GeometryAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in 3D Geometry
A regular pyramid S.ABCD (or S.ABC) can be rotated with the mouse or a finger. The simulation constructs the projection of the lateral edge SA (or the slant height SM) onto the base, shades the angle between the line and the plane, and shows tan of that angle in terms of the height and the base edge; it also constructs the distance from S to the base and from the center O to the lateral face (SBC). The 2D | 3D button switches to a true 3D pyramid (WebGL) showing the line, its projection onto the base, the shaded angle φ and the segment OH perpendicular to (SBC); drag to rotate, scroll or pinch to zoom.
Lesson: Lines perpendicular to a plane, angle between a line and a plane, distance from a point to a plane
What it shows
The angle between a line and a plane is the angle between the line and its orthogonal projection onto the plane. In a regular pyramid the apex S projects onto the centre O of the base, so the projection of the lateral edge SA is OA and the angle is SAO, with tan φ = h/OA; for the slant height SM the angle is SMO, with tan φ = h/OM, which is also the angle between face (SBC) and the base. Because BC is perpendicular to plane (SOM), the perpendicular OH from O to SM is also perpendicular to face (SBC), so d(O, (SBC)) = SO·OM/SM.
How to use
Choose the Pyramid base, set Base edge a and Height SO = h, and select the Line whose angle with the base is examined. Rotate the figure until the right triangle SOA or SOM is clearly visible. Increase h and predict how φ will change. Switch on Show the distance from O to the lateral face (SBC) to see OH, and use the 2D | 3D button for a true 3D view that can be rotated and zoomed.
Parameters you can change
- Pyramid base Square ABCD (regular quadrilateral pyramid), Equilateral triangle ABC (regular triangular pyramid)
- Base edge a 2–6
- Height SO = h 1–6
- Line whose angle with the base is examined Lateral edge SA, Slant height SM (M is the midpoint of BC)
- Show the distance from O to the lateral face (SBC)
Questions to explore
- Which is larger, the angle between SA and the base or the angle between SM and the base? Why?
- How do both angles change when the height h increases while the base edge a stays fixed?
- Why does the perpendicular from O to face (SBC) lie in the plane (SOM)?
