3.2.2 Volume & Surface Area

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

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