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Home < Algebra Starter Lessons < 3 Solving Linear Equations << Using Letters for Physical Quantities
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Role of Letters in formulas for Areas of Rectangles, Triangles and CirclesWelcome. In some formulas for areas and perimeters, letters denote and serve as shorthand for lengths of sides or heights. We can describe calculations with or without knowing the measures of the lengths until the last possible moment.
1.1 RectanglesRecall the area of a rectangle is given by its length times its width. We can write this as
This is a longhand description of the computation of the area of a rectangle. If you give me the values of the length and the width, I can compute the area.
The computation of the area of the rectangle can be rewritten with shorthand notation as follows. To introduce shorthand notation, we say the area A of a rectangle is given by its width W times its length L. Here, we use A as shorthand for the area of a rectangle, L as shorthand for its length and W as shorthand for its width. The formula (recipe) for calculating the area A of a rectangle can be written more briefly as
or
The shorthand notation takes less space to write than the word-only description. Read the symbols × and 路 as times or multiply. The symbols × and 路 are both shorthand codes for times or multiplication. The dot symbol 路 is preferably to the times symbol × when the latter could be confused with the letter x. Confusion can occur because the letter x which many people write is too similar to the multiplication symbol ×. Shorthand notation provides a code for the description of calculations. Formula decoding is required. The shorthand formula A = L 路W is more compact (takes less room) than the word-only description. This formula is meaningless for us when the role of the letters in this shorthand description is not explained. To understand and to use the shorthand description or formula, you need information. You need to know or find what numbers or quantities the symbols mean or represent. In the above formula, L stood for the length of a rectangle. This has to be said to you or you have to ask. To anyone without this information, the formula remains mysterious. Talking about and describing computations almost gives us the power to do them. In the area calculation, the area A is obtained from the recipe A = L×W provided the length L and W are given or can be found. Without this information, we can describe or understand a calculation but not use it. The above rectangle example reminds us of the following:
1.2 TrianglesIn words, the area of a triangle is given by one half the length of a base of the triangle multiplied by the height of the triangle. This formula can be justified but at this moment we will not worry about why it holds. We may also write more briefly
We may write still more briefly that the area of a triangle is given by
This involves some shorthand notation: the letters A, B and H. When you read or decode this shorthand notation, remember B stands for the length of a base of the triangle. Also remember H stands for the height of the triangle above this base. Lastly, remember A stands for the area of the triangle.
We have used single letters in this shorthand description of the calculation. Any mark or squiggle or symbol you can draw and name can serve as shorthand for some number or quantity. Perhaps, we should use Atriangle or another symbol, since we have already used the letter A in the previous rectangle example. Alternatively, we adopt the following rule: while you are reading this triangle example, we use the letter A here as shorthand for the area of the triangle only. More will be said on using and reusing (recycling) shorthand symbols (for example, letters) and the roles they take. Think of them as actors which can perform many parts. They may take only one role in any scene, except for stories and scenes involving identical twins or cases of mistaken identities. |
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Home < Algebra Starter Lessons < 3 Solving Linear Equations << Using Letters for Physical Quantities
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