Mr Barton’s Maths NotesShape and Space6. Volumewww.mrbartonmaths.com
6. VolumeThe Beauty of the PrismGood News: So long as you know what a prism is, and you remember how to work out the areas ofthose 6 shapes we talked about in the last section (5. Area), you can do pretty much any volumequestion 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 everyonethe exact same size piece!prismprismnot a prismnot a prismprismnot a prismprism
Working out the Volume of a PrismSo long as you can work out the area of the repeating face of the prism, the formula for thevolume is the same for every single one:Volume of a Prism = Area of Repeating Face x LengthExample 1 – Cuboid5 cm8 cm4 cmFACERectangleArea = b h×8 5× = 40cm2Area =Area of Repeating FaceVolume of Prism40 4×= 160cm3
Example 2 – Triangular Based Prism11 m6 m5 m15 mNote: Don’t think you must use every measurement they give you. The 15m turned out to bepretty useless to us!Area of Repeating FaceFACETriangleArea =2b h×Area =6211×= 33m2Volume of Prism33 5×= 165m3
Example 3 – Cylinder6.2 mm3 mmFACEArea of Repeating FaceCircleArea =2rπ ×Area == 28.274… mm29π= ×23π ×Note: Sometimes “length” can mean “height” when you are working out the volume of theprism. It just depends which way the repeating face is facing!Volume of Prism28.274... 6.2×= 175.3mm3(1dp)Note: Keep this value in your calculatorand use it for the next sum. It keepsyour answer nice and accurate!
Example 4 – Complicated Prism Note: This is still a prism as the front facerepeats throughout the object!FACEArea of Repeating FaceThis time it’s a bit more complicated as we cannot workout the area of the face in one go. We must first work outthe area of the complete rectangle, and then SUBTRACTthe area of the missing circle to get our answer!RectangleArea = b h×7 5×= 35m2Area =CircleArea =2rπ ×Area == 7.068… m22.25π= ×251.π ×Area of Repeating Face = 35 - 7.068…= 27.931…Volume of Prism27.931 3×= 83.8m3(1dp)7 m5 m3 m1.5 mNote: Try to avoid rounding in your workingout by keeping the big numbers in thecalculator, and then only round at the end!
Working out the Volume of Pointy ShapesObviously, not all 3D shapes have a repeating face. Some shapes start off with a flat face andend up at a point. The technical name I have given to these shapes is… Pointy Shapes!Volume of a Pointy Shape = Area of Face x LengthMore Good News: Just like prisms, there is a general rule for working out the volume of all shapeslike these:3
Example 4 – Cone180 m50 mArea of FaceCircleArea =2rπ ×Area == 25,446.9… m28100π= ×209π ×Volume of Pointy Shape25,446.9... 503×= 424,115 m3(nearest whole number)Note: Keep this value in your calculatorand use it for the next sum. It keepsyour answer nice and accurate!FACE Diameter = 180mRadius = 90 m
Example 5 – SphereSpheres do not have a repeating face, and they do not end in a pointy bit, so they have a ruleall to themselves, and here it is…r Volume of a Sphere =343rπ12 kmVolume of Sphere341203π× ×41,728,0003π= × ×37,238,229 km=