3D Shapes (Cambridge International IGCSE Mathematics)

Properties of 3D Shapes

What common 3D shapes do I need to know about?

• There are a number of common 3D shapes

  • You should know their names
  • and their key properties

• 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

  • There are other prisms, such as triangular prisms or hexagonal prisms

    • In these cases the exam question will make sure the shape of the cross-section is clear

  • A cylinder is similar to a prism

    • The cross-section of a cylinder is a circle

• There are a number of different types of pyramid

  • A pyramid has a flat base and sloping sides that meet at a point

  • The pyramid may have a square, rectangular or triangular base

  • Some pyramids have special names you should know

    • A triangular-based pyramid is called a tetrahedron

  • A cone is similar to a pyramid

    • A cone has a circular base

• A sphere is a round solid ball shape

What are the properties of 3D shapes?

• 3D shapes have a number of faces, vertices and edges

  • A face is a single flat 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 (and other surfaces) for the common 3D shapes

  • A cube has 6 equal, square faces

  • A cuboid has 3 pairs of rectangular faces

  • A cylinder has 2 equal circular faces and 1 curved surface

    • The curved surface has the shape of a rectangle if laid out flat

  • A triangular prism has 2 equal triangular faces and 3 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 1 square face and 4 equal triangular faces

  • A tetrahedron has 4 triangular faces

  • A sphere has 1 curved surface; it is a ball shape

Exam Hint Remembering the properties of 3D shapes will help in particular with questions involving surface area.

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
  • Solids can have more than one arrangement that will work to make the 3D shape

• The area of the net of a 3D shape is the same as the surface area of the solid

Nets of solids

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

  • Pay attention to which rectangles are the same

    • They are colour-coded in the diagram below

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

  • $\mathrm{Circumference}=2\timesπ\times\mathrm{radius}$

• The width of the rectangle is equal to the height of the cylinder

Net of a cylinder

What does the net of a pyramid look like?

• The net of a pyramid consists of the base and a triangle attached to each edge of the base

• For example, a square-based pyramid has a square base and four congruent (identical) triangles

  • The perpendicular height of each triangle is equal to the slant height of the pyramid

Example of a net of a pyramid

Exam Hint You may be given the dimensions of the solid when asked to draw a net

Make sure you put the correct lengths in the correct places by imagining cutting out and folding up the net.

Worked Example A cuboid measures 6 cm by 3 cm by 2 cm.

On the 1 cm² grid, draw an accurate net of this cuboid. One face has been drawn for you.

Answer:

The cuboid has three pairs of rectangles; measuring 6 cm by 3 cm, 6 cm by 2 cm, and 3 cm by 2 cm

Make sure the net has two of each of these rectangles in the correct places