Mr Fs Maths Notes Shape and Space 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: cm 2, mm 2, km 2, m 2, etc Three Dimensions (3D) Objects have a VOLUME Units of measurement include: cm 3, mm 3, km 3, m 3, etc Four Dimensions (4D) Objects exist in different times! Fortunately we dont 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 question Remember: pi (π) is just a number! 3. You should now be left with an expression just containing Ds, which you can use your algebra skills to simplify Crucial: When you are simplifying, DO NOT cancel anything out! Youll 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 D 2 - this is a formula for area D 3 - 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!

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 Okay, so our variables are w and h, and they become D 2. Lets get rid of our number 3. We only have Ds left in our expression, so its looking good! Now, lets use our algebra skills to simplify, remembering that in algebra the multiplication sign is disguised! 4. We are left with: Which means this is a formula for… AREA

2. 1. Okay, so our variables are w, l and h, and they become D 2. Lets get rid of our numbers 3. Now its time to simplify… but be careful! Its 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

3. 1. Okay, so our variables are w, l and h, and they become D 2. Lets get rid of our numbers… remember, pi (π) is just a number, and so are fractions! 3. Now its time to simplify… but be careful! Its fine to expand our brackets, but do not cancel anything out! Im 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 and

4. 1. Okay, so our variables are w, l and h, and they become D 2. Lets get rid of our numbers… 3. Now its 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

5. 1. Okay, so our variables are w, l and h, and they become D 2. Lets get rid of our numbers… and our constant K! 3. Now its time to simplify… Im 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

Good luck with your revision!