Volume (OCR GCSE Maths)

What is volume?

• The volume of a 3D shape is a measure of how much space it takes up
• You need to be able to calculate the volumes of a number of common 3D shapes

How do I find the volume of cuboids, prisms, and cylinders?

• To find the volume of a cuboid use the formula
  Volume of a cuboid = length × width × height

• • You will sometimes see the terms  'depth' or 'breadth' instead of 'height' or 'width'
  • A cuboid is in fact another name for a rectangular-based prism

• To find the volume of a prism use the formula
  Volume of a prism = area of cross-section × length

• • Note that the cross-section can be any shape, so as long as you know its area and length, you can calculate the volume of the prism
  • Or if you know the volume and length of the prism, you can calculate the cross-section area

• To calculate the volume of a cylinder with radius, r and height, h, use the formula
  Volume of a cylinder = πr²h

• • Note that a cylinder is in fact a circular-based prism: its cross-section is a circle with area πr², and its length is h

How do I find the volume of pyramids, cones, & spheres?

• To calculate the volume of a pyramid with height h, use the formula
  Volume of a pyramid = 1/3 × area of base × h

• • Note that to use this formula the height must be a line from the top of the pyramid that is perpendicular to the base

• To calculate the volume of a cone with base radius r and height h, use the formula
  Volume of a cone = 1/3 πr²h

• • Note that a cone is in fact a circular-based pyramid
  • As with a pyramid, to use the cone volume formula the height must be a line from the top of the cone that is perpendicular to the base

• To calculate the volume of a sphere with radius r, use the formula
  Volume of a sphere = 4/3 πr³

How can I solve problems when its not a "standard" 3D shape?

• Often the shape in a question will not be a standard cuboid, cone, sphere, etc
• It will likely either be:
  • A prism (3D shape with the same cross-section running through it)
  • A portion or fraction of a standard shape (a hemisphere for example)
• If the shape is a prism, recall that the volume of a prism is the cross sectional area × its length
  • The cross-sectional area may be a compound shape, such an an L-shape, or a combination of a rectangle and a triangle for example
• If the shape is a fraction of a standard shape, consider the "full" version of the shape and then find the appropriate fraction of it
  • A hemisphere is half a sphere
  • A frustum is a truncated (chopped-off) cone or pyramid
    • The volume of a frustum will be the volume of the smaller cone or pyramid subtracted from the volume of the larger cone or pyramid