3D Shapes (Cambridge O Level Maths)

You will need to remember the names and properties of many 3D shapes, the number of faces they have, the shapes of the faces and where they have planes of symmetry.

What 3D shapes should I know?

• You should know the general names of prisms
• A prism is a 3D shape with the same cross-section throughout
• The cross-section of a cube is a square
• The cross-section of a cuboid is a rectangle
• The cross-section of a cylinder is a circle
• You will also have to work with other prisms, such as triangular prisms
• In these cases the exam question will make sure the shape of the cross-section is clear
• You should know the names and properties of the different types of pyramids
• A pyramid has a flat base with sloping sides that meet at a point at the top
• The base will be a 2D shape
• Most pyramids are named after the shape of their base
• e.g. a square–based pyramid
• Some pyramids have special names you should know
• A cone has a circular base
• A tetrahedron’s base is an equilateral triangle
• You should know the name and properties of a sphere

What are the properties of 3D shapes?

• 3D shapes have a number of faces, vertices and edges
• A face is an individual surface of the 3D shape
• A vertex (plural, vertices) is a corner of the 3D shape
• An edge joins one vertex to another
• You should know the number and shape of the faces for the common 3D shapes
• A cube has 6 equal, square faces
• A cuboid has 3 pairs of equal, rectangular faces
• A cylinder has two equal circular faces and one rectangular face (its curved surface area)
• A triangular prism has two equal triangular faces and three rectangular faces
• If the triangular faces are equilateral then all of the rectangles will be equal
• If the triangular faces are isosceles then two of the rectangles will be equal
• A square-based pyramid has one square face and four equal triangular faces
• A tetrahedron has four triangular faces
• 3D shapes can be unfolded into a net to display their faces in 2D
• A net is just a pattern that could be folded to make the 3D shape

What is a net of a solid?

• A net of a solid is a 2D drawing that can be cut out and folded to make a 3D shape
• Each of the faces of the 3D shape are arranged in a certain pattern
  • Not every arrangement of the faces will create a net of that solid   
  • Many solids can have more than one different arrangement that will work to make the 3D shapes
• The area of the net of a 3D shape is the same as the surface area of the solid

What does the net of a cube or cuboid look like?

• The net of a cube has 6 squares connected at certain edges
• There are 11 different arrangements of the square faces that will form a net of a cube

• The most common and easiest to remember is in the form of a cross

• A cuboid has 6 rectangular faces, so its net consists of 6 rectangles
• The rectangles will be in three pairs of two 
• There are 54 different nets of a cuboid
  • Again, the most common and easiest to remember are in the form of a cross
  • Pay attention to which rectangles are the same

What does the net of a cylinder look like?

• The net of a cylinder consists of two circles and a rectangle
• The length of the rectangle is equal to the circumference of the circles
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• The width of the rectangle is equal to the height of the cylinder

What does the net of a pyramid look like?

• The net of a square based pyramid consists of one square and four congruent (identical) triangles
  • The side of the square is equal to the length of the base of each triangle
  • The perpendicular height of each triangle is equal to the slant height of the pyramid