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 3D 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
 Where A is the area of the cross section and h is the base height
 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 rightpyramid 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 rightpyramid 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 circularbased pyramid with the vertical height joining the apex to the centre of the circular base
 To calculate the volume of a rightcone 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
 Where r is the radius
Worked Example
A dessert can be modelled as a rightcone of radius 3 cm and height 10 cm and a scoop of icecream in the shape of a sphere of radius 3 cm. Find the total volume of the icecream and cone.
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 righttriangle 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

Worked Example
In the diagram below ABCD is the square base of a right pyramid with vertex V . The centre of the base is M. The sides of the square base are 3.6 cm and the vertical height is 8.2 cm.
i)
Use the Pythagorean Theorem to find the distance VN.
ii)
Calculate the area of the triangle ABV.
iii)
Find the surface area of the right pyramid.