Volumes Of Solids. : Volumes Of Solids. 14cm 5 cm 7cm 4cm 6cm 10cm 3cm 4cm What Is Volume ? : What Is Volume ? The volume of a solid is the amount of space inside the solid. Consider the cylinder below: If we were to fill the cylinder with water the volume would be the amount of water the cylinder could hold: Measuring Volume. : Measuring Volume. Volume is measured in cubic centimetres (also called centimetre cubed). Here is a cubic centimetre It is a cube which measures 1cm in all directions. We will now see how to calculate the volume of various shapes. Volumes Of Cuboids. : Volumes Of Cuboids. Look at the cuboid below: We must first calculate the area of the base of the cuboid: The base is a rectangle measuring 10cm by 3cm: Slide 5: Area of a rectangle = length x breadth Area = 10 x 3 Area = 30cm2 We now know we can place 30 centimetre squares on the base of the cuboid. But we can also place 30 cubic centimetres on the base: Slide 6: We have now got to find how many layers of 1cm cubes we can place in the cuboid: We can fit in 4 layers. Volume = 30 x 4 Volume = 120cm3 That means that we can place 120 of our cubes measuring a centimetre in all directions inside our cuboid. Slide 7: We have found that the volume of the cuboid is given by: Volume = 10 x 3 x 4 = 120cm3 This gives us our formula for the volume of a cuboid: Volume = Length x Breadth x Height V=LBH for short. What Goes In The Box ? : What Goes In The Box ? Calculate the volumes of the cuboids below: 490cm3 39.3cm3 76.9 m3 The Cross Sectional Area. : The Cross Sectional Area. When we calculated the volume of the cuboid : We found the area of the base : This is the Cross Sectional Area. The Cross section is the shape that is repeated throughout the volume. We then calculated how many layers of cross section made up the volume. This gives us a formula for calculating other volumes: Volume = Cross Sectional Area x Length. Slide 10: For the solids below identify the cross sectional area required for calculating the volume: The Volume Of A Cylinder. : The Volume Of A Cylinder. Consider the cylinder below: It has a height of 6cm . What is the size of the radius ? 2cm Volume = cross section x height What shape is the cross section? Circle Calculate the area of the circle: A = r 2 A = 3.14 x 2 x 2 A = 12.56 cm2 Calculate the volume: V = r 2 x h V = 12.56 x 6 V = 75.36 cm3 The formula for the volume of a cylinder is:
V = r 2 h
r = radius h = height. The Volume Of A Triangular Prism. : The Volume Of A Triangular Prism. Consider the triangular prism below: Volume = Cross Section x Height What shape is the cross section ? Triangle. Calculate the area of the triangle: A = ½ x base x height A = 0.5 x 5 x 5 A = 12.5cm2 Calculate the volume: Volume = Cross Section x Length V = 12.5 x 8 V = 100 cm3 The formula for the volume of a triangular prism is :
V = ½ b h l
B= base h = height l = length What Goes In The Box ? 2 : What Goes In The Box ? 2 Calculate the volume of the shapes below: 2813.4cm3 30m3 288cm3 More Complex Shapes. : More Complex Shapes. Calculate the volume of the shape below: Calculate the cross sectional area: Area = A1 + A2 Area = (12 x 16) + ( ½ x (20 –12) x 16) Area = 192 + 64 Area = 256m2 Calculate the volume: Volume = Cross sectional area x length. V = 256 x 23 V = 2888m3 Slide 15: Calculate the volume of the shape below: Calculate the cross sectional area: Area = A1 + A2 Area = (12 x 10) + ( ½ x x 6 x 6 ) Area = 120 +56.52 Area = 176.52cm2 Calculate the volume. Volume = cross sectional area x Length V = 176.52 x 18 V = 3177.36cm3 Example 2. What Goes In The Box ? 3 : What Goes In The Box ? 3 4466m3 19156.2cm3 Volume Of A Cone. : Volume Of A Cone. Consider the cylinder and cone shown below: The diameter (D) of the top of the cone and the cylinder are equal. The height (H) of the cone and the cylinder are equal. If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ? 3 times. This shows that the cylinder has three times the volume of a cone with the same height and radius. Slide 18: The experiment on the previous slide allows us to work out the formula for the volume of a cone: The formula for the volume of a cylinder is : V = r 2 h We have seen that the volume of a cylinder is three times more than that of a cone with the same diameter and height . The formula for the volume of a cone is: Slide 19: Calculate the volume of the cones below: Summary Of Volume Formula. : Summary Of Volume Formula.