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Gnvq key skills VOLUME Page 1 of 3 mps@WESTMINSTER COLLEGE Volume The space within an object is termed its volume or capacity. It is calculated by multiplying the objects three dimensions together (height, width and depth). The dimensions must be measured in the same units, i.e. they must all be in cm or metres or feet or inches etc.. Since the volume is three measurements multiplied together it is always measured in cubic measurements e.g. metres3 , feet3, miles3 or kilometres3 . You should find this process fairly easy for cubes and cuboids, but other shapes are slightly more difficult. Volume of Cubes and Cuboids. A cube is a three-dimensional shape with its sides the same length, and all its angles being 900. A cuboid is similar in that all its angles are 90 0 but its sides do not have to be the same length. Their volumes are calculated using the formula volume = w x d x h where w = width; d = depth; h = height. CUBE HEIGHT DEPTH CUBOID WIDTH (W) Example 1 1. A hall is 25 metres long, 17.6 metres wide and 3.15 metres high, what is its volume? The volume is the length x width x height which is 25 x 17.6 x 3.15 = 1386 m 3. 2. A box measures 3 feet long, 1 foot 3 inches wide and 2 foot 6 inches high, what is its volume? The dimensions have to be turned to feet from feet and inches, so they become 3 6 3 x 1 /12 x 2 /12 = 3 x 1.25 x 2.5 = 9.375 ft3. Exercise 1 1. A computer case has the following dimensions 58cm (l), 45cm (w) and 28cm (h). What is its volume? A cube has sides of 50cm. What is its volume in cm3 and m3? An organisation keeps all its files in rigid cardboard folders, each one 32cm by 25cm by 2.2cm. They need to store 1000 of these folders in a storeroom. How much space would they take up? A plank of wood is 2.8m long, 65 cm wide and 9cm high. What is its volume? 2. 3. 4. Gnvq key skills VOLUME Page 2 of 3 mps@WESTMINSTER COLLEGE Volume of Spheres and Cylinders A sphere is like a ball being circular in all its dimensions, while a cylinder is a circle in cross-section with one dimension being straight. d i a m e t e r cross-section h e i g h t radius The volume of a sphere is calculated using the formula volume = 4/3 r3. is a constant and has a value of roughly 3.14. r is the radius, which is the distance from the centre of the sphere to the outside. It is half the diameter, which is the distance across the sphere passing through the middle. r3 is the radius cubed, i.e. r x r x r. This is needed for Level 3 Application of Number The volume of a cylinder is calculated by first finding its cross-sectional area, (the area of its rounded face), and then multiplying by its height. The cross-sectional area is calculated by the formula area = r2. This means that the volume is = r2h, where h is the height. The same process can be applied to all cylinders including irregular shaped cylinders and prisms. As long as you know the cross-sectional area then the volume is the area multiplied by the height. This is needed for Level 2 Application of Number Example 2 1. A rubber ball inflates to a maximum diameter of 40cm, what is its volume? Volume = 4/3 r 3. The radius is 40/2 = 20cm, so its volume is 4 / 3 x 3.14 x 20 x 20 x 20 = 4/3 x 3.14 x 8000 = 31,400 cm 3 2. A water tank is a cylinder of 80 centimetres radius and height 10 meters, how much water can it hold? Volume is = r2h where the r is 80cm and h is 10m. The units must first be made the same. The radius can be changed to m (80/100 = 0.8m); or the height changed to cm (10x100 = 1000cm). Volume = 3.14 x 0.8 x 0.8 x 10 = 20.096 m3 or 3.14 x 80 x 80 x 1000 = 200,960 cm3 (See the section on converting volumes for more details on why the volume in cm 3 is so much larger than the volume in m3). 3. An irregular column has a cross-sectional area of 2.4m2 and a height of 4m, so its volume is 2.4 x 4 = 9.6m 2. Gnvq key skills VOLUME Page 3 of 3 mps@WESTMINSTER COLLEGE Exercise 2 1. A circular swimming pool is in the shape of a cylinder. If it has a diameter of 20 metres and a depth of 150 centimetres, how much water does it contain? 2. A ball has a radius of 30cm when inflated. How much air does it contain? 3. A tube has a diameter of 2cm and a length of 15m, what is its volume? 1.5m 4. A tent is in the shape of a hemisphere, (half a sphere), with a height of 1.5m. What is its volume? Converting Volumes To convert volumes of shapes you need to consider the fact that the conversion factor (if you have forgotten what this is refer to the sheets on converting measurements), is cubed. This means that to convert from cm3 to m3 you need to divide the value by 100 x 100 x 100 = 1,000,000, and to convert from m3 to cm3 you need to multiply the value by 1,000,000. To convert from ft3 to in3 you need to multiply by 12 x 12 x 12 = 1,728. To convert from in3 to ft3 you need to divide by 12 x 12 x 12 = 1,728. To convert from in3 to cm3 you need to multiply by 2.54 x 2.54 x 2.54 = 16.4 To convert from ft3 to m3 you need to multiply by 0.305 x 0.305 x 0.305 = 0.0284, or to divide by 3.283 x 3.283 x 3.283 = 35.38. The answer will be the same in both cases. Example 3 1. A box has a volume of 25cm3 and you need to convert this to mm3. 1 cm equals 10mm so 1cm3 is the same as 10 x 10 x 10 = 1,000mm3 . This means that 25cm3 will equal 25 x 1,000 = 25,000cm3. 2. A building has a volume of 25,000 feet3 and you need to convert this to metres3. 1 foot equals 0.305 metres, so 1 m3 is the same as 0.305 x 0.305 x 0.305 = 0.0284 ft 3. This means that 25,000 ft3 will equal 25,000 x 0.0284 = 710 m3 Exercise 3 1. A cylinder has a volume of 3.6m 3, what is its volume in cm3? 2. A box has a volume of 5.4 ft3, what is its volume in in3 ? 3. A building has a volume of 2,560 yards 3, what is this in metres 3 ? 4. A bottle contains 16in3 of liquid, how many bottles would be needed to fill a 500cm3 container? (Make the units the same and divide the bottles volume into the containers).