Is the area of a square proportional to the length of its sides?

The side length and area of a square are not proportional.

Students often misjudge the length of the sides of a square in proportion to its area. The reason for this miscalculation lies in the lack of a comprehensive understanding of the proportional relationship. The phrase "two related quantities, one quantity changes, and the other quantity also changes", is to remember, that the side length increases, and the area of the square also increases, but this is only half of the meaning of the proportional relationship.

Another sentence, which has been overlooked, is: "If the ratio of the two corresponding numbers in these two quantities is certain".

The ratio of the two numbers corresponding to the side length and area of the square is not equal. The ratio of any two values of the side length of a square to the corresponding area is not equal, so the side length of the square cannot be proportional to the area.

This is easy to understand.

Suppose the side length of the square is a, we know that the area of the square is a We find that there is a square relationship between the two, and the area increases with the increase of the side length, and it increases very quickly.

In addition, the definition of proportional relation, proportionality belongs to a primary function, that is, a primary function of y=kx+b,x,y, when k 0, it is a proportional relation, if k 0, it is an inverse proportional relation.

So this comparison shows that it is not proportional, but it is correct that the area increases with the length of the edge.

The side length of the square is proportional to the circumference. The area of the square changes with the change of the side length, but the ratio of the area of the square to the side length is not fixed, so the side length and area of the square are not proportional.

Proportional to the square, it is one of the special parallelograms. That is, a group of parallelograms with equal adjacent sides and one angle is at right angles is called a square, also known as a regular quadrilateral.

Square, with all the characteristics of a rectangle and a rhombus.

1. Proportional.

If the ratio of the two numbers corresponding to these two quantities (that is, the quotient) is constant, these two quantities are called proportional quantities, and their relationship is called proportional relations, and the proportional image is a straight line;

Expressed by letters is that if the letters x and y are used to represent two related quantities, and k is used to represent their ratio (certainly), the proportional relationship can be expressed by the following relation: y x=k (certainly);

2. Inverse proportionality.

If the product of the two numbers corresponding to these two quantities is constant, these two quantities are called inversely proportional quantities, and their relationship is called inversely proportional relations;

Methods for judging whether two quantities are proportional, inversely proportional, or disproportionate:

1) Find out the two quantities that are correlated.

2) List the quantity relation according to the relationship between the two related quantities.

3) If the ratio (i.e., quotient) of the two corresponding numbers in the two quantities is certain, it is a proportional quantity; If the product is fixed, it is an inversely proportional amount.

Not. Proportional.

This is because the ratio of area to side length is not a fixed value (constant).

Sorry, disproportionate, it should be that the circumference of the square is proportional to the length of the sides.

The area of the square and the side length are not proportional, the reason is: the relationship between the area of the square and the side length is: the area of the square is equal to the side length The side length, if the side length is expanded n times, the area of the square will be expanded by n square times, such as the side length of the square is 5, the area is 5 2 = 25, if the area is expanded by two times, the side length is 10, the area is 10 2 = 100, and it is expanded by 4 times, and the proportion of the two changes is not the same, so the area of the square and the side length are not proportional.

The meaning of proportionality is: one number expands n times, and the other number also expands n times together, and the value of the two numbers expands is the same, then the two numbers are said to be proportional.

Square area formulaThe formula for the area of a square is the length of the side of the side.

There is a parallelogram with right angles and equal adjacent edges.

It's a square. Nature of the square:

1. The two groups of opposite sides are parallel to each other, the four sides are equal, and the adjacent sides are perpendicular to each other.

2. The four corners are 90°, and the sum of the inner angles is 360°.

3. Diagonal.

perpendicular to each other; The diagonals are equal and bisected from each other; Each diagonal is divided into a set of diagonals.

4. It is a center-symmetrical figure.

Again, axisymmetric figures (there are four axes of symmetry.

The area of a square is disproportionate to the length of its sides for the following reasons: whether it is proportional or inversely proportional, there must be a certain amount of it (or invariant).

Because one of the characteristics of the square is that all four sides of the square are of equal length. The area of the square = the length of the side The length of the side = the length of the side

In the above formula, a certain amount cannot be found, if one side length expands, the other side lengths must also expand accordingly, otherwise it is not a square. So, the side length and area of the square are disproportionate.

At the same time, it should also be noted that although the length and area of the sides of the square are not proportional, the square and area of the sides of the square are directly proportional. Because the ratio of the square of the side length and the two numbers corresponding to the area is equal.

1. Proportional.

If the ratio of the two numbers corresponding to these two quantities (that is, the quotient) is constant, these two quantities are called proportional quantities, and their relationship is called proportional relations, and the proportional image is a straight line;

Expressed by letters is that if the letters x and y are used to represent two related quantities, and k is used to represent their ratio (certainly), the proportional relationship can be expressed by the following relation: y x=k (certainly);

2. Inverse proportionality.

If the product of the two numbers corresponding to these two quantities is constant, these two quantities are called inversely proportional quantities, and their relationship is called inversely proportional relations;

Nature of the square:

1. The two groups of opposite sides are parallel to each other; All four sides are equal; Adjacent edges are perpendicular to each other.

2. The four corners are 90°, and the sum of the inner angles is 360°.

3. The diagonals are perpendicular to each other; The diagonals are equal and bisected from each other; Each diagonal is divided into a set of diagonals.

4. It is both a center-symmetrical figure and an axisymmetric figure (with four axes of symmetry).

5. A diagonal line of the square divides the square into two congruent isosceles right-angled triangles, and the angle between the diagonal and the side is 45°; The two diagonal lines of the square divide the square into four congruent isosceles right triangles.

6. The square has all the properties and characteristics of a parallelogram, a rhombus, and a rectangle.

7. Draw the largest circle (the inscribed circle of the square) in the square, and the area of the circle is about one-tenth of the area of the square]; The area of the smallest circle that completely covers the square (the circumscribed circle of the square) is about 157% of the area of the square [2/2].

8. The square is a special rectangle, and the square is a special diamond.

According to the meaning of proportionality and inverse proportionality, in the proportional quantity relationship, there is a certain quantity, two changing quantities, if the three quantities are changing, then there is a disproportional relationship Solution: the area of the square = the length of the side The length of the side, when the side length of the square changes, the other side of it also changes, and the area also changes at the same time, and these three quantities are all changing, so the area of the square is not proportional to the length of the side

The side length and area of a square are not proportional.

Because: square area = side length side length, that is: square area side length = side length, if the side length is fixed, the area must not change.

So, the side length of the square is (not) proportional to the area.

s=a is not proportional (linear) relation.

It is a curvilinear (parabolic) relationship.

Because: side length x side length = square area (definitely).

So: the length of the side is inversely proportional to the area (relationship).

It is not proportional to the relationship, and the side length must be certain.

Disproportionately, the area will change when the side length changes, such as: 3 3 9, 4 4 16. Area Edge length Edge length (the value of the edge length is not necessarily).

It can be judged in the form of **.

The area of a square is not proportional to the length of its sides. To determine the ratio between two related quantities, it depends on whether the ratio of the two quantities is a certain or a certain product. If the ratio is constant, it is proportional; If the product is constant, it is inversely proportional.

Because 1 1 = 1, 4 2 = 2 9 3 = 3, 16 4 = 4, that is, the ratio of the area of the square to the length of the sides is not fixed.

The main features of the square

1. Edges: two groups of opposite sides are parallel to each other; All four sides are equal; Adjacent edges are perpendicular to each other.

2. Inner angles: All four corners are right angles.

3. Diagonal: the diagonals are perpendicular to each other; The diagonals are equal and bisected from each other; Each diagonal is divided into a set of diagonals. The diagonal lines are equal.

4. Symmetry: It is both a central symmetrical figure and an axisymmetric figure (with four axes of symmetry).

5. The square has all the properties of a parallelogram, a rhombus, and a rectangle.

6. Special properties: a diagonal line of the square divides the square into two congruent isosceles right triangles, and the angle between the diagonal and the side is 45°; The two diagonal lines of the square divide the square into four congruent isosceles right triangles.

7. Draw the largest circle in the square, and the area of the circle is about the area of the square; The circumscribed circle area of the square is about 157% of the area of the square.

8. The square is a special rectangle.

9. The midpoint quadrilateral of the square is a square, and the ratio of the area is 1:2.

The side length and area of the square are not proportional to the reason: because the area of the square changes with the change of the side length, but the ratio of the area of the square to the length of its side is not a fixed value, which does not meet the condition of proportional relationship, so the side length of the square is disproportionate to its area.

For the square, the square is a special parallelogram, its four corners are all right angles, and the length of the four sides is the same, according to the formula of the circumference of the square: c=4a, it can be seen that the side length and perimeter of the square are proportional to each other.

According to the meaning of positive proportion and inverse proportionality, in the proportional quantity relationship, there is a certain quantity, two changing quantities, if the three carrying amounts are all changing, then there is a disproportional relationship.

When the length of the side of the square changes, the other side of the square also changes, and the area also changes at the same time, and these three quantities are all changed, so the area of the square is not proportional to the side length.