Transcript This is just a test to see if notes will appear here…

Maths Notes
Shape and Space
7. Dimensions
www.mrbartonmaths.com
7. Dimensions
What are Dimensions?
You may have heard people taking about dimensions in terms of objects:
One Dimension (1D)
Objects have just a LENGTH
Units of measurement include:
cm, mm, km, m, mile, etc
Two Dimensions (2D)
Objects have an AREA
Units of measurement include:
cm2, mm2, km2, m2, etc
Three Dimensions (3D)
Objects have a VOLUME
Units of measurement include:
cm3, mm3, km3, m3, etc
Four Dimensions (4D)
Objects exist in different times!
Fortunately we don’t need to worry about this!
The advantage of knowing this is that when we are given a formula, we can tell whether it is
one for LENGTH, AREA, VOLUME, or just a load of rubbish!
Using Dimensions to Discover what Formulas are actually Working Out
Again, this is just my way of doing this, and feel free to bin it if you have a better one!
1. Change all the variables in the formula to the letter D
Note: Variables are just letters that represent lengths, widths and heights
2. Ignore all numbers (apart from powers!) and constants
Note: If a letter represents a constant instead of a variable, it will well you in the question
Remember: pi (π) is just a number!
3. You should now be left with an expression just containing D’s, which you can use your algebra
skills to simplify
Crucial: When you are simplifying, DO NOT cancel anything out! You’ll see why in the examples!
4. Look at what you are left with. If the formula only contains…
D - this is a formula for length
D2 - this is a formula for area
D3 - this is a formula for volume
Any combination - this formula is rubbish!
Examples
In all the following examples, l, w and h are variables representing lengths, and k is a
constant
Determine whether these formulas calculate length, area, volume or nothing
1.
5wh
1. Okay, so our variables are w and h, and they become D
2. Let’s get rid of our number
3. We only have D’s left in our expression, so it’s looking good! Now,
let’s use our algebra skills to simplify, remembering that in algebra
the multiplication sign is disguised!
4. We are left with:
D
2
Which means this is a formula for… AREA
5DD
DD
D2
2.
7h(l  w)  2w
2
1. Okay, so our variables are w, l and h, and they become D
D( D  D )  D 2
2. Let’s get rid of our numbers
3. Now it’s time to simplify… but be careful! It’s fine to expand our
brackets, but do not cancel anything out!
4. We are left with a formula that just contains:
Which means this is a formula for… AREA
7 D ( D  D)  2 D 2
D2
D2  D2  D2
2 h(lh   w  h 2 )
3
3.
1. Okay, so our variables are w, l and h, and they become D
2 D( DD   D  D 2 )
3
2. Let’s get rid of our numbers… remember, pi (π) is just a number,
and so are fractions!
D( DD  D  D2 )
3. Now it’s time to simplify… but be careful! It’s fine to expand our
brackets, but do not cancel anything out! I’m going to do this in two
stages!
D( D 2  D  D 2 )
4. We are left with a formula that contains a mixture of:
D
2
and
D
3
Which means this formula is a load of rubbish
D3  D 2  D3
4.
5h3  2lw2  hlw
6
1. Okay, so our variables are w, l and h, and they become D
D3  DD2  DDD
2. Let’s get rid of our numbers…
3. Now it’s time to simplify… but be careful! We are definitely not
going to cancel anything out!
4. We are left with a formula that only contains:
Which means this formula is for volume
5D3  2 DD 2  DDD
6
D3
D3  D3  D3
5.
kl 3   hw2
8hl
1. Okay, so our variables are w, l and h, and they become D
kD3   DD 2
8DD
D3  DD 2
DD
2. Let’s get rid of our numbers… and our constant K!
3. Now it’s time to simplify… I’m going to simplify the terms on the
top and bottom first, and then divide the top by the bottom!
D3  D3
D2
DD
4. We are left with a formula that only contains:
Which means this formula is for length
D
Good luck with
your revision!