1. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 5.5 Volume and surface area

2. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area The following presentation looks at how to calculate the volume and surface area of common three-dimensional shapes. To find the surface area, it is sometimes easier to visualise the shape as a net. Cube l l l Volume = l × l × l = l 3 l l Area of one square face = l × l = l 2 Area of six faces = 6l 2

3. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area A possible net is shown below. The area of each face is shown opposite. Cuboid l w h Volume = l × w × h = lwh l l w h h h lw lh wh Total surface area = 2wh + 2lw + 2lh = 2(wh + lw + lh)

4. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area Total surface area = sum of areas of each face Triangular prism b h l x Volume = area of cross-section × length = ½ × b × h × l = ½bhl l b h x x To find the value of x use Pythagoras’ rule on the triangular face. Therefore

5. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area Length of rectangle = circumference of circle = 2πr Cylinder r h h r Volume = area of cross-section × height = π × r 2 × h = πr 2 h Surface area = 2πr 2 + 2πrh = 2πr (r + h)

6. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area Cone r l h Volume = Surface area = area of base + area of curved face = πr 2 + πr l = πr (r + l)

7. Mathematical Studies for the IB Diploma Second Edition © Hodder & Stoughton Ltd 2012 Volume and surface area Sphere r Volume = Surface area = 4πr 2