Volume

Defn Volume The volume of a solid is the amount of space it occupies. In other words, it is the number of cubes which fit inside. Volume is measured in either cm3 or m3 Here is a cubic centimetre: It is a cube which measures 1cm in all directions. 1cm

Defn Polyhedron Polyhedron: A solid three dimensional shape, with flat faces e.g:

Defn Cuboid: A cuboid, is a polyhedron with six rectangular plane faces E.g:

Defn: Prism A prism is a solid with two opposite faces that are the same shape, size and are parallel. I.e. it has the same cross-section for the entire length of the shape. http://www.mathsisfun.com/geometry/prisms.html This web site has a few examples of prisms

The following are all prisms Cylinder Cuboid Cross section Triangular Prism Trapezoid Prism Volume of Prism = length x Cross-sectional area

Volumes 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: 3cm 10cm

3cm 10cm 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: 10cm 3cm 4cm

10cm 3cm 4cm 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.

The Volume of a prism is simply the area of one end times the length of the prism

10cm 3cm 4cm 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.

Do we need to see that again? How to find volume 3x6 = 18 cubes 3x6 = 18 cubes 18 cubes x 4 lots = 72 cubes

Find the volume of the following? Calculate the volumes of the cuboids below: (1) 14cm 5 cm 7cm (2) 3.4cm 490cm3 39.3cm3 (3) 8.9 m 2.7m 3.2m 76.9 m3

The Cone + + = Volume of a Cone = A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + = Volume of a Cone =

Identical isosceles triangles Pyramids A Pyramid is a three dimensional figure with a regular polygon as its base and all the outside faces are identical isosceles triangles meeting at a point. Identical isosceles triangles base = quadrilateral base = heptagon base = pentagon

= + + Volume of Pyramids Volume of a Pyramid: V = (1/3) Area of the base x height V = (1/3) Ah Volume of a Pyramid = 1/3 x Volume of a Prism = + +

The Cross Sectional Area. When we calculated the volume of the cuboid : 10cm 3cm 4cm 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.

For the solids below identify the cross sectional area required for calculating the volume: (2) (1) Right Angled Triangle. Circle (4) (3) A2 A1 Rectangle & Semi Circle. Pentagon

The Volume Of A Cylinder. Consider the cylinder below: It has a height of 6cm . 4cm 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 The formula for the volume of a cylinder is: V =  r 2 h r = radius h = height. Calculate the volume: V =  r 2 x h V = 12.56 x 6 V = 75.36 cm3

The Volume Of A Triangular Prism. Consider the triangular prism below: 5cm 8cm 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 The formula for the volume of a triangular prism is : V = ½ b h l B= base h = height l = length V = 12.5 x 8 V = 100 cm3

What Goes In The Box ? 2 Calculate the volume of the shapes below: (2) (1) 16cm 14cm (3) 6cm 12cm 8m 2813.4cm3 30m3 288cm3

More Complex Shapes. Calculate the volume of the shape below: 20m 23m Volume = Cross sectional area x length. V = 256 x 23 A1 A2 V = 2888m3 Calculate the cross sectional area: Area = A1 + A2 Area = (12 x 16) + ( ½ x (20 –12) x 16) Area = 192 + 64 Area = 256m2

Example 2. Calculate the volume of the shape below: 12cm 18cm 10cm A2 A1 Calculate the volume. Volume = cross sectional area x Length V = 176.52 x 18 V = 3177.36cm3 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

What Goes In The Box ? 3 11m (1) 4466m3 14m 22m (2) 18m 17cm

Other slides:

Volumes Of Solids. 8m 5m 7cm 5 cm 14cm 6cm 4cm 4cm 3cm 10cm

What Is Volume ? The volume of a solid is the amount of space inside the solid. In other words it is the number of cubes which fit inside. 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. 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. 1cm We will now see how to calculate the volume of various shapes.

Volumes 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: 3cm 10cm

3cm 10cm 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: 10cm 3cm 4cm

10cm 3cm 4cm 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.

10cm 3cm 4cm 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 ? Calculate the volumes of the cuboids below: (1) 14cm 5 cm 7cm (2) 3.4cm 490cm3 39.3cm3 (3) 8.9 m 2.7m 3.2m 76.9 m3

The Cross Sectional Area. When we calculated the volume of the cuboid : 10cm 3cm 4cm 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.

For the solids below identify the cross sectional area required for calculating the volume: (2) (1) Right Angled Triangle. Circle (4) (3) A2 A1 Rectangle & Semi Circle. Pentagon

The Volume Of A Cylinder. Consider the cylinder below: It has a height of 6cm . 4cm 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 The formula for the volume of a cylinder is: V =  r 2 h r = radius h = height. Calculate the volume: V =  r 2 x h V = 12.56 x 6 V = 75.36 cm3

The Volume Of A Triangular Prism. Consider the triangular prism below: 5cm 8cm 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 The formula for the volume of a triangular prism is : V = ½ b h l B= base h = height l = length V = 12.5 x 8 V = 100 cm3

What Goes In The Box ? 2 Calculate the volume of the shapes below: (2) (1) 16cm 14cm (3) 6cm 12cm 8m 2813.4cm3 30m3 288cm3

More Complex Shapes. Calculate the volume of the shape below: 20m 23m Volume = Cross sectional area x length. V = 256 x 23 A1 A2 V = 2888m3 Calculate the cross sectional area: Area = A1 + A2 Area = (12 x 16) + ( ½ x (20 –12) x 16) Area = 192 + 64 Area = 256m2

Example 2. Calculate the volume of the shape below: 12cm 18cm 10cm A2 A1 Calculate the volume. Volume = cross sectional area x Length V = 176.52 x 18 V = 3177.36cm3 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

What Goes In The Box ? 3 11m (1) 4466m3 14m 22m (2) 18m 17cm

Volume 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. H 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 ? This shows that the cylinder has three times the volume of a cone with the same height and radius. 3 times.

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: h r r = radius h = height

Calculate the volume of the cones below: (2) 9m 6m (1)

Summary Of Volume Formula. h V =  r 2 h l b h V = l b h b l h V = ½ b h l h r

The Cone + + = Volume of a Cone = A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + = Volume of a Cone =

Exercise #1 = (1/3)(3.14)(1)2(2) = 3.14(1)2(2) = 2.09 m3 = 6.28 m3 Find the volume of a cylinder with a radius r=1 m and height h=2 m. Find the volume of a cone with a radius r=1 m and height h=1 m Volume of a Cylinder = base x height = pr2h = 3.14(1)2(2) = 6.28 m3 Volume of a Cone = (1/3) pr2h = (1/3)(3.14)(1)2(2) = 2.09 m3

Find the area of a cone with a radius r=3 m and height h=4 m. Surface Area of a Cone Find the area of a cone with a radius r=3 m and height h=4 m. r = the radius h = the height l = the slant height Use the Pythagorean Theorem to find l l 2 = r2 + h2 l 2= (3)2 + (4)2 l 2= 25 l = 5 Surface Area of a Cone = pr2 + prl = 3.14(3)2 + 3.14(3)(5) = 75.36 m2

Identical isosceles triangles Pyramids A Pyramid is a three dimensional figure with a regular polygon as its base and lateral faces are identical isosceles triangles meeting at a point. Identical isosceles triangles base = quadrilateral base = heptagon base = pentagon

= + + Volume of Pyramids Volume of a Pyramid: V = (1/3) Area of the base x height V = (1/3) Ah Volume of a Pyramid = 1/3 x Volume of a Prism = + +

Exercise #2 Find the volume of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m Volume = 1/3 (area of base) (height) = 1/3 ( 60m2)(8m) = 160 m3 h Area of base = ½ Pa a = ½ (5)(6)(4) = 60 m2 s

Area of Pyramids Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m Surface Area = area of base + 5 (area of one lateral face) What shape is the base? Area of a pentagon h = ½ Pa = ½ (5)(6)(4) = 60 m2 l a s

Area of Pyramids h l a s What shape are the lateral sides? Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m Area of a triangle = ½ base (height) = ½ (6)(8.9) = 26.7 m2 Attention! the height of the triangle is the slant height ”l ” h l l 2 = h2 + a2 = 82 + 42 = 80 m2 l = 8.9 m a s

Area of Pyramids h l a s Surface Area of the Pyramid = 60 m2 + 5(26.7) m2 = 60 m2 + 133.5 m2 = 193.5 m2 Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m h l a s

A Prism Cross section Volume of Prism = length x Cross-sectional area Cylinder Cuboid Cross section Triangular Prism Trapezoid Prism Volume of Prism = length x Cross-sectional area

Area Formulae r h b Area Circle = π x r2 Area Rectangle = Base x height h b h b Area Trapezium = ½ x (a + b) x h a b h Area Triangle = ½ x Base x height

Volume Cylinder Cross-sectional Area = π x r2 = π x 32 = 28.2743…..cm2 DO NOT ROUND! 3cm 5cm USE CALCULATOR ‘ANS’! Volume = length x CSA = 5 x 28.2743…. = 141.3716….cm3 = 141.4cm3

Volume Cuboid Cross-sectional Area = b x h = 7.2 x 5.3 = 38.16cm2 DO NOT ROUND! 7.2cm Volume = length x CSA USE ‘ANS’! = 10.6 x 38.16 = 404.496cm3 Sensible degree of accuracy = 404.5cm3

Volume Trapezoid Prism Cross-sectional Area = ½ x(a + b) x h = ½ x (6.3 + 1.7) x 4.9 1.7cm 8.2cm 6.3cm 4.9cm = 19.6cm2 DO NOT ROUND! Volume = length x CSA USE ‘ANS’! = 19.6 x 8.2 = 160.72cm3 Sensible degree of accuracy = 160.7cm3

Volume Triangular Prism Cross-sectional Area = ½ x b x h = ½ x 8.6 x 4.1 = 17.63cm2 4.1cm 4.9cm DO NOT ROUND! 6.2cm 8.6cm Volume = length x CSA USE ‘ANS’! = 17.63 x 6.2 = 109.306cm3 Sensible degree of accuracy = 109.3cm3