The number of rows is the height of the rectangle.
The number of squares in one row is the length of the rectangle. Similarly, has 3 rows of 7 squares (or 7 columns of 3 squares), for a total of 7 × 3 squares, so its area is 21 square units. We can count the two rows of seven squares. If we decide that the area of this square is 1, then a rectangle that is 7 times as long would have 7 × 1 as its area.Ī rectangle that is twice the height of would have twice its area, so the area of is 2 × 7 units of area. (See also surface area.) Area of rectanglesīy choosing a square as the unit of area, we get an intuitive idea about the area of rectangles. Each formula they reinvent helps strengthen their understanding (and memory) for the other formulas they know. For now, we will just use this as a fact.Students who have the informal notion that area is the “amount of 2-D ‘stuff'” contained inside a region can invent for themselves most of the formulas that they are often asked merely to memorize. You can see this most easily when you draw a parallelogram on graph paper or look at the diagram below. In high school, you will be able to prove that a perpendicular segment from a point on one side of a parallelogram to the opposite side will always have the same length. This is so that we don’t get confused about whether × means multiply, or whether the letter x is standing in for a number.
Notice that the multiplication symbol can be written with a small dot instead of a × symbol. If b is base of a parallelogram (in units), and h is the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers b Even though the two rectangles have different side lengths, the products of the side lengths are equal, so they have the same area! And both rectangles have the same area as the parallelogram. Notice that the side lengths of each rectangle are the base and height of the parallelogram. There are infinitely many line segments that can represent the height! If we draw any perpendicular segment from a point on the base to the opposite side of the parallelogram, that segment will always have the same length.Both the side (the segment) and its length (the measurement) are called the base. We can choose any of the four sides of a parallelogram as the base.If both the height and base were 100 times the original the area would be 100 × 100 = 10000 times the original area. If both the height and base tripled, the area would be 3 × 3 = 9 times the original area. Hence, if a given height h and a given base b are doubled the result would be 2 b × 2 h = 4 A, where A was the original area. If the height is 100 times the original, the area would be 100 times the original. If the height triples, the area would triple. Hence, if a given height h doubles the result would be b × 2 h = 2 A, where A was the original area.
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