Volume | Mathematics for GCSE/IGCSE - Year 11 PDF Download

What is volume?

• The volume of a three-dimensional shape indicates the amount of space it occupies.
• It's important to know how to compute the volumes of several common three-dimensional 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.

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Volume Calculation for Prisms

• Formula for Volume: To determine the volume of a prism, you can use the formula where 
• Volume of a prism = area of cross-section × length = Al

• Varied Cross-Sections: A prism's cross-section can take any shape. If you know its area and length, you can calculate the volume of the prism.
• Reverse Calculation: Alternatively, if you're aware of the volume and length of the prism, you can determine the cross-section's area.

Volume of a Cylinder

• To find the volume of a cylinder with radius (r) and height (h), apply the formula: Volume of a cylinder = πr²h

• It's important to note that a cylinder essentially behaves like a circular-based prism, where its cross-section is a circle with an area of πr² and a length of h.

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

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

• It's crucial to understand that for this formula, the height must be a line from the apex of the pyramid that is perpendicular to the base.

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Volume of a Cone

• Volume of a Cone: To find the volume of a cone with base radius 'r' and height 'h', use the formula Volume of a cone =

• Cone Structure: Note that a cone is essentially a circular-based pyramid. Like in pyramids, the height in the cone volume formula should be a line from the apex perpendicular to the base.

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Volume of a Sphere

• Volume of a Sphere: Calculate the volume of a sphere with radius 'r' using the formula Volume of a sphere =

Problem Solving with Volumes

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

• Non-standard shapes in problems often deviate from typical forms like cubes, cones, or spheres.
• They typically fall into two categories:
  • A prism, which is a 3D shape with a consistent cross-section throughout its length.
  • A portion or fraction of a standard shape, such as a hemisphere.
• If the shape is a prism:
  • Remember that the volume of a prism equals the cross-sectional area multiplied by its length.
  • The cross-sectional area might be a complex shape like an L-shape or a combination of a rectangle and a triangle.
• If the shape is a fraction of a standard shape:
  • Consider the complete version of the shape and then calculate the appropriate fraction of it.
  • For example, a hemisphere is half of a sphere, and a frustum is a truncated cone or pyramid.
  • The volume of a frustum is found by subtracting the volume of the smaller shape from the volume of the larger shape.