Geometry of 3D Shapes

Geometry of 3D Shapes

Understanding 3D Shapes

  • Familiarise yourself with the common 3D shapes: cube, cuboid, cylinder, cone, sphere, pyramid, and prism.
  • Learn the formulas for the volume and surface area of these shapes and be able to apply them.
  • Understand how to identify each of these unique shapes based on their properties: edges, vertices, and faces.


  • Comprehend the definition and characteristics of polyhedra, which are 3D shapes with flat faces.
  • A polyhedron has a definite number of faces, edges, and vertices. Familiarise yourself with how these quantities relate to each other (Euler’s formula: Faces + Vertices - Edges = 2).
  • Identify the difference between regular and irregular polyhedra. Regular polyhedra, also known as Platonic solids, have faces that are regular polygons of the same size and shape.

Surface Area and Volume

  • Recognize the surface area of a 3D shape as the total area of all its outside surfaces.
  • Know how to calculate the surface area of a shape, typically by finding the area of each face and adding them together.
  • Understand volume as the quantity of 3D space enclosed by a closed surface.
  • Learn and apply the formulas for finding the volume of a shape.

Sphere, Cylinder and Cone

  • Realize that the formulas are different when it comes to a sphere, cylinder, and cone due to their curved surfaces.
  • For a sphere, memorise and understand how to apply the formulas for surface area (4πr²) and volume (4/3πr³).
  • With a cylinder, be able to calculate the surface area (2πr(r + h)) and volume (πr²h) accurately.
  • Whereas for a cone, familiarise yourself with the formulas for surface area (πr(r + l)) and volume (1/3πr²h), where l represents the slant height.

Cross Sections

  • Understand the concept of cross sections, which are the shapes you get when you slice through a 3D shape.
  • Be able to identify the cross section of different 3D shapes, and how the position of the slice can change the resulting cross section.
  • Know the difference between horizontal and vertical cross sections.

Practical Applications

  • Apply all these principles to solve real-world problems that involve 3D shapes — for instance, in packaging, construction or design-related challenges.
  • Combine 3D shape understanding with other mathematical concepts, such as ratios or algebra, for complex problem-solving.