Planar section
SŠ13 minSlide and tilt the cutting plane and watch the shape of the section change.
Solid
With two points, A and B are opposite corners of the rectangular base. Every further point widens the base until it encloses that point too. The base sits at the lowest of the given heights and the body rises by h.
Points · 2
max 12Further points widen the beam until it encloses them. For a base with any number of vertices, switch to Prism.
Planar section
The section is a 4-gon.
3D view
Perspective
Bring the eye closer and parallel edges rush together toward a vanishing point; move it away and perspective tends to parallel projection.
Measurements
- Volume
- 16.50
- Surface
- 39.20
- Vertices V
- 8
- Edges E
- 12
- Faces F
- 6
- V − E + F
- 2
V − E + F = 2 holds for every simple polyhedron (Euler’s formula) – a quick check that the body is closed.
Theory
A planar section of a solid is a polygon. Its vertices are exactly the points where the cutting plane pierces the solid’s edges – so the whole task decomposes into a series of line–plane intersections you already know how to construct.
The number of section vertices tells you how many edges the plane crosses. That is why the shape changes as you slide or tilt the plane: a beam gives a quadrilateral, but a plane slicing through a corner can give a triangle or a pentagon.
A useful check: the section lies entirely in one plane, so its vertices must be coplanar, and consecutive vertices always lie on adjacent faces. If a face gets skipped, one intersection is missing.
Step-by-step construction
Practice
Build the construction yourself
What is the next step?
1 / 3Choose the step that comes next. The drawing only advances on a correct choice.
Check your understanding
What is the planar section of a polyhedron?
What are the vertices of the section?
Why does the number of section sides change as you tilt the plane?
Downloadable material
A PDF for this lesson is in preparation.
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