Presentation on theme: "Volumes Of Solids. 8m 5m 7cm 5 cm 14cm 6cm 4cm 4cm 3cm 10cm."— Presentation transcript:
1Volumes Of Solids.8m5m7cm5 cm14cm6cm4cm4cm3cm10cm
2What 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:
3Measuring Volume.Volume is measured in cubic centimeters (also called centimeter cubed).Here is a cubic centimeterIt is a cube which measures 1cm in all directions.1cmWe will now see how to calculate the volume of various shapes.
4Volumes Of Cuboids. Look at the cuboid below: 4cm 3cm 10cm We must first calculate the area of the base of the cuboid:The base is a rectangle measuring 10cm by 3cm:3cm10cm
53cm10cmArea of a rectangle = length x widthArea = 10 x 3Area = 30cm2We now know we can place 30 centimeter squares on the base of the cuboid. But we can also place 30 cubic centimeters on the base:10cm3cm4cm
610cm3cm4cmWe 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 4Volume = 120cm3That means that we can place 120 of our cubes measuring a centimeter in all directions inside our cuboid.
710cm3cm4cmWe have found that the volume of the cuboid is given by:Volume = 10 x 3 x 4 = 120cm3This gives us our formula for the volume of a cuboid:Volume = Length x Width x HeightV=LWH for short.
8What Goes In The Box ? Calculate the volumes of the cuboids below: (1) 14cm5 cm7cm(2)3.4cm490cm339.3cm3(3)8.9 m2.7m3.2m76.9 m3
9The Cross Sectional Area. When we calculated the volume of the cuboid :10cm3cm4cmWe 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.
10For the solids below identify the cross sectional area required for calculating the volume: (2)(1)Right Angled Triangle.Circle(4)(3)A2A1Rectangle & Semi Circle.Pentagon
11The Volume Of A Cylinder. Consider the cylinder below:It has a height of 6cm .4cm6cmWhat is the size of the radius ?2cmVolume = cross section x heightWhat shape is the cross section?CircleCalculate the area of the circle:A = r 2A = 3.14 x 2 x 2A = cm2The formula for the volume of a cylinder is:V = r 2 hr = radius h = height.Calculate the volume:V = r 2 x hV = x 6V = cm3
12The Volume Of A Triangular Prism. Consider the triangular prism below:5cm8cmVolume = Cross Section x HeightWhat shape is the cross section ?Triangle.Calculate the area of the triangle:A = ½ x base x heightA = 0.5 x 5 x 5A = 12.5cm2Calculate the volume:Volume = Cross Section x LengthThe formula for the volume of a triangular prism is :V = ½ b h lB= base h = height l = lengthV = 12.5 x 8V = 100 cm3
13What Goes In The Box ? 2 Calculate the volume of the shapes below: (2) (1)16cm14cm(3)6cm12cm8m2813.4cm330m3288cm3
14More Complex Shapes. Calculate the volume of the shape below: 20m 23m Volume = Cross sectional area x length.V = 256 x 23A1A2V = 2888m3Calculate the cross sectional area:Area = A1 + A2Area = (12 x 16) + ( ½ x (20 –12) x 16)Area =Area = 256m2
15Example 2.Calculate the volume of the shape below:12cm18cm10cmA2A1Calculate the volume.Volume = cross sectional area x LengthV = x 18V = cm3Calculate the cross sectional area:Area = A1 + A2Area = (12 x 10) + ( ½ x x 6 x 6 )Area =Area = cm2
16What Goes In The Box ? 3 11m (1) 4466m3 14m 22m (2) 18m 17cm
17Volume Of A Cone. Consider the cylinder and cone shown below: D The diameter (D) of the top of the cone and the cylinder are equal.HThe 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 ?This shows that the cylinder has three times the volume of a cone with the same height and radius.3 times.
18The 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 hWe 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:hrr = radius h = height
19Calculate the volume of the cones below: (2)9m6m(1)
20Summary Of Volume Formula. hV = r 2 hlbhV = l w hblhV = ½ b h lhr