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Dimension Theory A one-off presentation which, if understood and remembered, will skill you to answer that elusive GCSE question if it happens to appear this year! Example Lisa knows that the formula for the volume of a cylinder is either V = 2rh or V=r2h She also knows that r and h have dimensions of length whilst 2 and are numbers with no dimensions. Explain which of the formulae Lisa should use to work out the volume of a cylinder. LINES A line has only one dimension: the LENGTH. Length is measured using: mm or cm or m or km AREAS An area has two dimensions: we usually think of length x width but we could say LENGTH x LENGTH Area is measured using: mm² or cm² or m² or km² VOLUMES A volume has three dimensions: we usually think of length x width x height but we could say LENGTH x LENGTH x LENGTH Volume is measured using: mm³ or cm³ or m³ or km³ Dimensions of LENGTH, AREA and VOLUME. The dimension of perimeter is length (L) The dimension of area is length x length (L²) The dimension of volume is length x length x length (L³) Lisa knows that the formula for the volume of a cylinder is either V = 2rh or V=r²h She also knows that r and h have dimensions of length whilst 2 and are numbers with no dimensions. Explain which of the formulae Lisa should use to work out the volume of a cylinder. V = 2rh has dimension of V = number x number x length x length so V = length x length which is wrong. V=r²h has dimension of V = number x length² x length = number x length x length x length = length x length x length which is correct. Since volume has dimension length x length x length, Lisa should use V = r²h The expressions shown below can be used to calculate lengths, areas or volumes of various shapes. The letters r and h represent length. , 2, 3, 4, 5 and 10 are numbers which have no dimension. r(+2) 4r² h r(r+4h) rh 4 10r³ (r+2h) 3r³ h r²(h+r) Which of these can be used to calculate an area? 4r³ 5 Area has dimension length x length. This is what I am looking for in each of the given expressions: r(+2) has dimension length x number so it has dimensions of length. (Can’t be used.) 4r² has dimension number x length² x number. h length so it has dimensions of length. (Can’t be used.) r(r+4h) has dimension length x length. rh has dimension length x length so it has dimensions 4 number of length x length. 4r³ has dimension number x length³ so it has dimensions 5 number of length x length x length. (Can’t be used.) 10r³ has dimension number x length³ x number so it has dimensions of length x length x length. (Can’t be used.) (r+2h) has dimension number x length so it has dimensions of length. (Can’t be used.) 3r³ has dimension number x length³ so it has dimensions h length of length x length. r²(h + r) has dimension length² x length so it has dimensions of length x length x length. (Can’t be used.) So the expressions for area with dimension length x length are: r(r+4h) rh 4 3r³ h