Mr F’s Maths Notes

Shape and Space

7. Dimensions

7. Dimensions

What are Dimensions?

You may have heard people taking about dimensions in terms of objects:

One Dimension (1D)Objects have just a LENGTHUnits of measurement include: cm, mm, km, m, mile, etc

Two Dimensions (2D)Objects have an AREAUnits of measurement include: cm2, mm2, km2, m2, etc

Three Dimensions (3D)Objects have a VOLUMEUnits 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!

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 questionRemember: 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!

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!

ExamplesIn all the following examples, l, w and h are variables representing lengths, and k is a constantDetermine whether these formulas calculate length, area, volume or nothing

5wh1.

1. Okay, so our variables are w and h, and they become D 5DD

2. Let’s get rid of our number DD

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!

2D

4. We are left with:

Which means this is a formula for… AREA

2D

27 ( ) 2h l w w 2.

1. Okay, so our variables are w, l and h, and they become D

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

2D

27 ( ) 2D D D D

2( )D D D D

2 2 2D D D

22 ( )3 h lh w h 3.

1. Okay, so our variables are w, l and h, and they become D

2. Let’s get rid of our numbers… remember, pi (π) is just a number, and so are fractions!

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!

4. We are left with a formula that contains a mixture of:

Which means this formula is a load of rubbish

2D

22 ( )3 D DD D D

2( )D DD D D

2 2( )D D D D 3 2 3D D D

3Dand

3 25 2

6

h lw hlw 4.

1. Okay, so our variables are w, l and h, and they become D

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

3D

3 25 2

6

D DD DDD

3 2D DD DDD

3 3 3D D D

3 2

8

kl hw

hl

5.

1. Okay, so our variables are w, l and h, and they become D

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!

4. We are left with a formula that only contains:

Which means this formula is for length

D

3 2

8

kD DD

DD

3 2D DD

DD

3 3

2

D D

D

D D

Good luck with your revision!