Solid Geometry · GeoSpace

Solids

A solid bounded by plane polygons is a polyhedron; a solid swept by turning a plane figure about a line is a solid of revolution.

Drag the stage to turn the solid. Edges you cannot see are drawn dashed: the standard convention for drawing solids.

Count the vertices V, edges E and faces F of a polyhedron: they always satisfy Euler's formula V − E + F = 2.

Nets

Cut the surface along some edges and lay it flat to get a net. The surface area is the area of the net.

A cylinder's lateral surface unrolls into a rectangle, a cone's into a sector. A cube has 11 different nets.

For the shortest path between two points on the surface, unfold the faces it crosses into one plane and draw a straight line.

Views & oblique drawing

Looking from the front, the left and above gives the front, side and top views. They line up: equal length, equal height, equal width.

Click the grid to stack cubes; right-click (or switch to Remove) to take them away. The three views update as you build. Different stacks can share the same three views.

Oblique drawing shows a horizontal plane figure in perspective.

Cross section

Cutting a solid with a plane gives a plane figure called a cross section. Each side of it is where the cutting plane meets one face of the solid.

Drag the three points P, Q, R along the edges to set the plane, or switch to Free plane to set its direction and position.

A cube's section can be a triangle, quadrilateral, pentagon or hexagon, but a triangular section is always acute, and a regular pentagon is impossible.

Angles in space

Every angle in space is found as a plane angle:

Skew lines: translate one line until the two meet; the acute or right angle between them, in (0°, 90°].

Line and plane: the angle between the line and its projection onto the plane, in [0°, 90°].

Dihedral angle: from a point on the edge, draw a ray perpendicular to the edge in each face; the angle between the rays, in [0°, 180°].

Click edges, diagonals or faces to pick objects, or choose an example.

Vectors & coordinates

With a coordinate system, points have coordinates, lines have direction vectors and planes have normal vectors. Angles and distances follow from vector arithmetic, with no construction lines.

Skew lines: cos θ = |a·b| / (|a||b|); line and plane: sin θ = |a·n| / (|a||n|); dihedral angle: from the angle between normals, reading acute or obtuse off the figure; point to plane: d = |AP·n| / |n|.

The panel sets the vector result beside the geometric method from chapter 5.

Circumscribed & inscribed spheres

A sphere through all the vertices of a solid is its circumscribed sphere; a sphere touching all its faces is its inscribed sphere.

The key is the center, equidistant from every vertex. For a cuboid, the space diagonal is a diameter.

A regular tetrahedron, or a tetrahedron with three mutually perpendicular edges, can be completed to a cube or cuboid with the same circumscribed sphere.

In-class testing

Five random questions on cross sections, nets and three views. After you answer, the stage shows the solution.