Volumes, Surface Areas and Nets

Volume is the space occupied by a 3-D shape. It is calculated by multiplying the three dimensions together. Consider this Cuboid h w d The volume is V = w x d x h Units will be……. Unit 3 such as cm 3, m 3, etc.

Before we go any further… A word or two about volume units If a length is in cm, then an area (if measured in cm) is… cm x cm = cm 2 This means than an equivalent shape’s volume is… cm x cm x cm = cm 3

Converting cm 3 to mm 3 and m 3 to cm 3 1 cm = 10 mm (check your ruler) So if cm 3 = cm x cm x cm Then 1 cm 3 = 10 mm x 10 mm x 10 mm = 1000 mm 3 Likewise 1 m = 100 cm So 1 m 3 = 100 cm x 100 cm x 100cm = cm 3

IF, (and this is a big IF) the shape is made from smaller cubes, you can count the cubes, remembering all those that are hidden. This shape is made from 1cm cubes. There are 9 cubes on an exposed face. The cube is also 3 cubes deep. (3 layers of 3 cubes) So the volume will be 3 x 9 = 27 cm 3

The Cylinder h r V = π x r 2 x h This is the area of the end (circle) multiplied by the height. This basic idea will help remember the volume of any PRISM shape !!!!

Prism Area of end cross section d Volume is V = cross section area x d

Other more complex shapes are combinations of those given. You will need to break the shapes down into simpler basic shapes. Work out the area of the “red” part then multiply by the depth (d) d

Work out the volume of these two shapes Diameter = 20 cm 60 cm 120 cm Cross sectional area 106cm cm cm 3

Surface area This is the area of all the surfaces of a particular shape. Cubes and cuboids have 6 surfaces and the area of each is calculated and added together. h w d S.A. = (2 x d x h) + (2 x d x w) + (2 x w x h) The “2” because opposite sides are equal in shape.

Consider a cube of side 2 cm It has 6 identical faces Each face has an area of 2cm x 2cm Which is 4 cm 2 Each surface of a cube is identical in area, so the total surface area of the cube is 6 x 4cm 2 Which is 24 cm 2 2 cm

It has six faces, opposite faces are identical. 2 faces are 2 x 3 cm = 2 x 6 cm 2 = 12 cm 2 2 faces are 3 x 4 cm = 2 x 12 cm 2 = 24 cm 2 2 faces are 2 x 4 cm = 2 x 8 cm 2 = 16 cm 2 Adding these areas together we get a surface area of = 52 cm 2 4 cm 2 cm 3 cm Now consider a cuboid of side 2 x 3 x 4 cm

Now consider a triangular prism The shape is made up of two triangles and three rectangles. The area of each triangle = ½ (3 x 4) = 6 cm All units are in cm Areas of the rectangles = (5 x 8) + ( 3 x 8) + ( 4 x 8) = = 96 cm 2 So the total surface area = (2 x 6) + 96 = 108 cm 2 You may need to use Pythagoras to work out one of the sides of the triangle. Remember that the area of a triangle is ½ x base x height

More complex shapes can be broken down into simpler shapes for calculation. However a cylinder is calculated as follows. h r SA = (2 x π r 2 ) + (2 π r h) The two circles The perimeter times the height of cylinder (the “unpeeled” body) next slide

Cylinder The “unpeeled” body This is also the “NET” of this shape.

If you unfold a 3D shape and lay it flat the shape will be the “net” of the 3D object. A cuboid Net of the cuboid You can also use this method to work out surface area. Just work out the areas of each individual rectangle and add them up.

Cylinder Net of a cylinder (As previously seen)

Match the shape and its net…. A > 3 B > 5 C > 7 D > 6 E > 1 F > 2 G > 4 Some of these we have already dealt with.