Transcript Dimension Theory

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 = 2rh
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 = 2rh
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 = 2rh 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