Volume & Surface Area (DP IB Analysis & Approaches (AA)): Revision Note

Volume of 3D Shapes

What is volume?

• The volume of a 3D shape is a measure of how much 3D space it takes up

  • A 3D shape is also called a solid

• You need to be able to calculate the volume of a number of common shapes

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

• A prism is a 3-D shape that has two identical base shapes connected by parallel edges

  • A prism has the same base shape all the way through

  • A prism takes its name from its base

• To find the volume of any prism use the formula:
  

  Volume of a prism = Ah
  

  • Where A is the area of the cross section and h is the base height

    • h could also be the length of the prism, depending on how it is oriented

  • This is in the formula booklet in the prior learning section at the beginning

  • The base could be any shape so as long as you know its area and length you can calculate the volume of any prism

• Note two special cases:

  • To find the volume of a cuboid use the formula:

    
    
    

  • The volume of a cylinder can be found in the same way as a prism using the formula:
    
    
    

  • where is the radius, is the height (or length, depending on the orientation

  • Note that a cylinder is technically not a prism as its base is not a polygon, however the method for finding its volume is the same

• Both of these are in the formula booklet in the prior learning section

How do I find the volume of pyramids and cones?

• In a right-pyramid the apex (the joining point of the triangular faces) is vertically above the centre of the base

  • The base can be any shape but is usually a square, rectangle or triangle

• To calculate the volume of a right-pyramid use the formula

  

  • Where A is the area of the base, h is the height

  • Note that the height must be vertical to the base

• A right cone is a circular-based pyramid with the vertical height joining the apex to the centre of the circular base

• To calculate the volume of a right-cone use the formula

  

  • Where is the radius, is the height

• These formulae are both given in the formula booklet

How do I find the volume of a sphere?

• To calculate the volume of a sphere use the formula
  
  
  

  • Where r is the radius

    • the line segment from the centre of the sphere to the surface

  • This formula is given in the formula booklet

Surface Area of 3D Shapes

What is surface area?

• The surface area of a 3D shape is the sum of the areas of all the faces that make up a shape

  • A face is one of the flat or curved surfaces that make up a 3D shape

  • It often helps to consider a 3D shape in the form of its 2D net

How do I find the surface area of cuboids, pyramids and prisms?

• Any prisms and pyramids that have polygons as their bases have only flat faces

  • The surface area is simply found by adding up the areas of these flat faces

  • Drawing a 2D net will help to see which faces the 3D shape is made up of

How do I find the surface area of cylinders, cones and spheres?

• Cones, cylinders and spheres all have curved faces so it is not always as easy to see their shape

  • The net of a cylinder is made up of two identical circles and a rectangle

  • The rectangle is the curved surface area and is harder to identify

  • The length of the rectangle is the same as the circumference of the circle

  • The area of the curved surface area is
    

    
    

    • where r is the radius, h is the height

  • This is given in the formula book in the prior learning section

  • The area of the total surface area of a cylinder is
    

    
    

  • This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given

• The net of a cone consists of the circular base along with the curved surface area

  • The area of the curved surface area is

    

    • Where r is the radius and l is the slant height

  • This is given in the formula book

    • Be careful not to confuse the slant height, l, with the vertical height, h

    • Note that r, h and l will create a right-triangle with l as the hypotenuse

  • The area of the total surface area of a cone is
    

    
    

  • This is not given in the formula book, however it is easy to put together as both the area of a circle and the area of the curved surface area are given

• To find the surface area of a sphere use the formula
  

  

  • where r is the radius (line segment from the centre to the surface)

  • This is given in the formula booklet, you do not have to remember it