AREAS - GEOMETRY.

CONCEPT OF AREA

It would be convenient for you to study this course in a different manner than the usual.

Try not to recite a formula. Instead, try to __understand and know what you are saying__ from the start. This implies a minor supplementary effort which I consider to be very important.

Up to now, we have studied points, lines, angles, polygons, etc. in geometry. We have represented them over a plane. Most of all, we have worked with lines; some curved and some straight, but lines nonetheless. In many exercises, we have only worked on calculating lengths.

From this point on, when studying areas, besides the __length__, we will need to know another dimension: __width__ or __height__ . Moreover, you need to study this topic using a __ruler, a pen, some scissors and paper.__

15(2).1 On a sheet of paper, draw a square measuring 1 cm. on its sides. Now, add 9 equal squares, as you can see in the image below. __Cut it__ and now you have a piece of paper 10 cm. long by 1 cm. wide in your hands:

How many 1 cm. squares do you have?

Answer: 10 squares measuring 1 cm. on each side.

15(2).2 Draw a line 10 cm. long and another line 5 cm. wide. Cut it. How many 1 cm. squares do you have?

Answer: 50 squares measuring 1 cm. on each side.

Solution:

15(2).3 Which is the easiest way of knowing the quantity of 1 cm. squares for the previous image?

Answer: Multiplying the number of squares we have length-wise (10) by the number of squares we have side-wise (5):

15(2).4 You have cut a piece of paper that measures 10 cm. long by 5 cm. wide. If you want to have a piece of paper measuring 8 cm. long by 10 cm. wide, is it necessary to draw 80 squares measuring 1 cm. on each side?

Answer: No, you simply need to draw a line 8 cm. long and another line 10 cm. wide.

15(2).5 Is this piece of paper measuring 8 cm. long by 10 cm. wide the same as the following image?

Answer:

CALCULATING THE AREA OF A RECTANGLE

When solving an exercise, whenever possible, you must write the answer with the type of unit used in the exercise. For example: 8 Km, 12 Ha, 4 m2., 12 cm., etc

On the previous image, you have a rectangle measuring 8 cm. long by 10 cm. wide. How many square centimetres do you have?

According to what we've been studying, we have:

Remember, when multiplying powers with the same base, we must add their exponents; cm has an exponent of 1, so: is the same as:

15(2).6 In the following figure, How many square centimetres do we have?

Answer:

Solution:

The number of square centimetres is 70 if you count them.

15(2).7 Using a ruler, draw a rectangle measuring 90 millimetres long and 50 millimetres wide. Cut it and test the answer. How many square millimetres does the rectangle have?

Answer: and the shape is:

15(2).8 How many can you find on a rectangle measuring 4 dm. wide by 3 dm. long?

Answer:

AREA AND SURFACE

These are words we will use indifferently; they will be the same. Some say the area is the numerical expression obtained after measuring a surface, others say similar things; we will use them indifferently.

All the pieces of paper you have cut have different sizes. As you have seen, to know the exact area of a surface, we need to know __two__ measurement: length and width (or height; imagine a wall measuring 3 metres long and has a height of 5 metres).

15(2).9 What is the area of a rectangular plaza measuring 90 metres long by 80 metres wide?

Answer: