Key Properties of Three-Dimensional Shapes

Three-dimensional shapes are all around us, from buildings to everyday objects. Understanding their properties, like faces, edges, and volume, helps us grasp the fundamentals of geometry and how these shapes fit into our world.

1. Cube

   • A cube has six equal square faces, twelve edges, and eight vertices.
   • All angles in a cube are right angles (90 degrees).
   • The volume of a cube is calculated as V = s³, where s is the length of a side.

2. Rectangular Prism

   • A rectangular prism has six faces, all of which are rectangles.
   • It has twelve edges and eight vertices, similar to a cube but with varying face dimensions.
   • The volume is calculated using V = l × w × h, where l is length, w is width, and h is height.

3. Sphere

   • A sphere is a perfectly round three-dimensional shape with all points on its surface equidistant from its center.
   • It has no edges or vertices.
   • The volume of a sphere is given by V = (4/3)πr³, where r is the radius.

4. Cylinder

   • A cylinder has two parallel circular bases connected by a curved surface.
   • It has two edges (the circular edges of the bases) and no vertices.
   • The volume is calculated as V = πr²h, where r is the radius of the base and h is the height.

5. Cone

   • A cone has a circular base and a single vertex (the apex) that is not in the same plane as the base.
   • It has one edge (the circular edge of the base) and one vertex.
   • The volume is calculated using V = (1/3)πr²h, where r is the radius of the base and h is the height.

6. Pyramid

   • A pyramid has a polygonal base and triangular faces that converge at a single vertex (the apex).
   • The number of triangular faces equals the number of sides on the base.
   • The volume is calculated as V = (1/3)Bh, where B is the area of the base and h is the height from the base to the apex.

7. Prism

   • A prism has two parallel bases that are congruent polygons and rectangular lateral faces.
   • The number of faces equals the number of sides on the base plus two.
   • The volume is calculated as V = Bh, where B is the area of the base and h is the height.

8. Polyhedron

   • A polyhedron is a three-dimensional shape with flat polygonal faces, straight edges, and vertices.
   • The most common types include prisms and pyramids.
   • The Euler's formula relates the number of vertices (V), edges (E), and faces (F) as V - E + F = 2.

9. Tetrahedron

   • A tetrahedron is a polyhedron with four triangular faces, six edges, and four vertices.
   • It is the simplest form of a three-dimensional shape.
   • The volume is calculated as V = (1/3)Bh, where B is the area of the triangular base and h is the height from the base to the apex.

10. Octahedron

    • An octahedron has eight triangular faces, twelve edges, and six vertices.
    • It can be visualized as two pyramids with square bases joined at their bases.
    • The volume is calculated as V = (1/3)Bh, where B is the area of the base and h is the height from the base to the apex.