Maths Notes Shape and Space 6. Volume www.mrbartonmaths.com

6. Volume The Beauty of the Prism What is a Prism? prism prism Good News: So long as you know what a prism is, and you remember how to work out the areas of those 6 shapes we talked about in the last section (5. Area), you can do pretty much any volume question without needing any more formulas!... But remember your answers are UNITS CUBED! What is a Prism? A Prism is a 3D object whose face is the exact same shape throughout the object. A birthday cake is the shape of a prism if it is possible to cut it in such a way to give everyone the exact same size piece! prism prism not a prism prism prism not a prism not a prism

Volume of a Prism = Area of Repeating Face x Length Working out the Volume of a Prism So long as you can work out the area of the repeating face of the prism, the formula for the volume is the same for every single one: Volume of a Prism = Area of Repeating Face x Length Example 1 – Cuboid Area of Repeating Face Rectangle Area = Area = = 40cm2 FACE 5 cm 4 cm Volume of Prism 8 cm = 160cm3

FACE Example 2 – Triangular Based Prism Area of Repeating Face Area = Triangle 15 m 11 m Area = FACE 5 m 6 m Area = = 33m2 Volume of Prism = 165m3 Note: Don’t think you must use every measurement they give you. The 15m turned out to be pretty useless to us!

FACE Example 3 – Cylinder Area of Repeating Face Area = Area = 3 mm Area of Repeating Face FACE Circle Area = 6.2 mm Area = = 28.274… mm2 Volume of Prism Note: Keep this value in your calculator and use it for the next sum. It keeps your answer nice and accurate! = 175.3mm3 (1dp) Note: Sometimes “length” can mean “height” when you are working out the volume of the prism. It just depends which way the repeating face is facing!

FACE Example 4 – Complicated Prism Area of Repeating Face Area = Note: This is still a prism as the front face repeats throughout the object! Area of Repeating Face This time it’s a bit more complicated as we cannot work out the area of the face in one go. We must first work out the area of the complete rectangle, and then SUBTRACT the area of the missing circle to get our answer! 7 m 1.5 m Rectangle Circle FACE Area = Area = 3 m Area = 5 m Area = = 35m2 Area of Repeating Face = 35 - 7.068… = 27.931… = 7.068… m2 Volume of Prism Note: Try to avoid rounding in your working out by keeping the big numbers in the calculator, and then only round at the end! = 83.8m3 (1dp)

Volume of a Pointy Shape = Area of Face x Length Working out the Volume of Pointy Shapes Obviously, not all 3D shapes have a repeating face. Some shapes start off with a flat face and end up at a point. The technical name I have given to these shapes is… Pointy Shapes! More Good News: Just like prisms, there is a general rule for working out the volume of all shapes like these: Volume of a Pointy Shape = Area of Face x Length 3

FACE Example 4 – Cone Area of Face Area = Area = = 25,446.9… m2 Circle FACE Area = Diameter = 180m Radius = 90 m 180 m Area = = 25,446.9… m2 Volume of Pointy Shape Note: Keep this value in your calculator and use it for the next sum. It keeps your answer nice and accurate! = 424,115 m3 (nearest whole number)

Volume of a Sphere = Example 5 – Sphere Volume of Sphere Spheres do not have a repeating face, and they do not end in a pointy bit, so they have a rule all to themselves, and here it is… Volume of a Sphere = r Volume of Sphere 12 km

Good luck with your revision!