How to find the area of sq. We calculate the area of a square: along the side, diagonal, perimeter

Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures of the Euclidean plane and satisfying 4 conditions:

Positivity - Area cannot be less than zero;
Normalization - a square with side unit has area 1;
Congruence - congruent figures have equal area;
Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.

Formulas for the area of geometric figures.

Geometric figure	Formula	Drawing
The result of adding the distances between the midpoints opposite sides of a convex quadrilateral will be equal to its semi-perimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half its radius.
Circle segment. To obtain the area of segment ASB, it is enough to subtract the area of triangle AOB from the area of sector AOB.	S = 1 / 2 R(s - AC)
The area of the ellipse is equal to the product of the lengths of the major and minor semi-axes of the ellipse and the number pi.
Ellipse. Another option for calculating the area of an ellipse is through two of its radii.
Triangle. Through the base and height. Formula for the area of a circle using its radius and diameter.
Square . Through his side. The area of a square is equal to the square of the length of its side.
Square. Through its diagonals. The area of a square is equal to half the square of the length of its diagonal.
Regular polygon. To determine the area of a regular polygon, you need to divide it into equal triangles, which would have a common vertex at the center of the inscribed circle.	S= r p = 1/2 r n a

Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited closed loop of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

Formula for the area of a triangle by side and height
Area of a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
Formula for the area of a triangle based on three sides and the radius of the circumcircle
Formula for the area of a triangle based on three sides and the radius of the inscribed circle
Area of a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.

Square area formulas

Formula for the area of a square by side length
Square area equal to the square of the length of its side.
Formula for the area of a square along the diagonal length
Square area equal to half the square of the length of its diagonal.
S= 1 2
2

Rectangle area formula

Area of a rectangle

Parallelogram area formulas

Formula for the area of a parallelogram based on side length and height
Area of a parallelogram
Formula for the area of a parallelogram based on two sides and the angle between them
Area of a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
a b sin α

Formulas for the area of a rhombus

Formula for the area of a rhombus based on side length and height
Area of a rhombus equal to the product of the length of its side and the length of the height lowered to this side.
Formula for the area of a rhombus based on side length and angle
Area of a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
Formula for the area of a rhombus based on the lengths of its diagonals
Area of a rhombus equal to half the product of the lengths of its diagonals.

Trapezoid area formulas

Heron's formula for trapezoid
Where S is the area of the trapezoid,
- lengths of the bases of the trapezoid,
- lengths of the sides of the trapezoid,

Area of a polygon

We will associate the concept of area of a polygon with this geometric figure like a square. For the unit area of a polygon we will take the area of a square with a side equal to one. Let us introduce two basic properties for the concept of area of a polygon.

Property 1: For equal polygons, their areas are equal.

Property 2: Any polygon can be divided into several polygons. In this case, the area of the original polygon is equal to the sum of the areas of all the polygons into which this polygon is divided.

Square area

Theorem 1

The area of a square is defined as the square of the length of its side.

where $a$ is the length of the side of the square.

Proof.

To prove this we need to consider three cases.

The theorem has been proven.

Area of a rectangle

Theorem 2

The area of a rectangle is determined by the product of the lengths of its adjacent sides.

Mathematically this can be written as follows

Proof.

Let us be given a rectangle $ABCD$ with $AB=b,\ AD=a$. Let's build it up to a square $APRV$, the side length of which is equal to $a+b$ (Fig. 3).

Figure 3.

By the second property of areas we have

\ \ \

By Theorem 1

\ \

The theorem has been proven.

Sample tasks

Example 1

Find the area of a rectangle with sides $5$ and $3$.

The area of a square is the part of the plane that is limited by the sides of this square.

A square is a special case of a rectangle, its area can be found as the product of one of its sides by the other, and since all sides of a square are equal, its area will be equal to the square of the length of its side:

Also, the area of a square is equal to half the square of the length of its diagonal (d), that is:

The diameter of a circle circumscribed about a square coincides with the diagonal of this square, then its area can be found through the length of the diameter (D) of the circumscribed circle:

Since the diameter of a circle is 2 times greater than its radius, the area of the square can also be found through the radius of the circumscribed circle:

S = (2 * R)²/2 = (4 * R²)/2 = 2 * R².

A square is a regular quadrilateral, that is, a quadrilateral in which all sides are equal. The area of a square can be found in three ways:

Through the side of the square.
Through the perimeter of the square.
Through the diagonal of the square.

Let's consider each of the methods for finding the area of a square.

Calculating the area of a square using its side

Let a be the side of the square. Since all sides of a square are equal, each side of the square will be equal to a. In this case, the area of the square S can be calculated using the formula:
S = a * a = a 2 . For example, let the side of a square be 5, then its area will be:
S = 5 2 = 25.

Calculating the area of a square using its perimeter

Let P be the perimeter of the square. The perimeter is the sum of all sides, then P = a + a + a + a = 4 * a. Since S = a 2 (according to the previously written formula), then a can be expressed from the perimeter:
a = P / 4. Then S = P 2 / 16. For example, it is known that the perimeter of a square is 20, then you can find its area: S = 20 2 / 16 = 400 / 16 = 25.

Calculating the area of a square using its diagonal

The diagonal of a square divides it into two equal right triangles. Consider one of the right triangles. Its legs are equal to a and a (two sides of the square), and the hypotenuse is equal to the diagonal of the square (d). Using the Pythagorean theorem, we calculate the hypotenuse:
d 2 = a 2 + a 2 ;
d 2 = 2 * a 2 ;
d = a * √2.
In this case, the area of the square will be written as follows: S = d 2 /2. For example, given the diagonal of a square: d = √18, then the area of the square will be: S = (√18) 2 / 2 = 18 / 2 = 9.
All these formulas are convenient for calculating the area of a square.

Square is a regular quadrilateral in which all angles and sides are equal to each other.

Quite often this figure is considered as special case or . The diagonals of a square are equal to each other and are used in the formula for the area of a square through the diagonal.
To calculate the area, consider the formula for the area of a square using the diagonals:

That is, the area of the square is equal to the square of the length of the diagonal divided by two. Given that the sides of the figure are equal, you can calculate the length of the diagonal from the area formula right triangle or by the Pythagorean theorem.

Let's look at an example of calculating the area of a square using the diagonal. Let a square with a diagonal d = 3 cm be given. It is necessary to calculate its area:

Using this example of calculating the area of a square using diagonals, we got the result 4.5 .

Area of a square by side

You can also find the area of a regular quadrilateral by its side. The formula for the area of a square is very simple:

Since in the previous example of calculating the area of a square we calculated the value by diameter, now let’s try to find the length of the side:
Let's substitute the value into the expression:
The side length of the square will be 2.1 cm.

You can very simply use the formula for the area of a square inscribed in a circle.

The diameter of the circumscribed circle will be equal to the diameter of the square. Since a square is considered a regular rhombus, you can use the formula for calculating the area of a rhombus. It is equal to half the product of its diagonals. The diagonals of the square are equal, so the formula will look like this:
Let's consider an example of calculating the area of a square inscribed in a circle.

Given a square inscribed in a circle. The diagonal of the circle is d = 6 cm. Find the area of the square.
We remember that the diagonal of a circle is equal to the diagonal of a square. We substitute the value into the formula for calculating the area of a square through its diagonals:

The area of the square is 18

Area of a square through perimeter

In some problems, the conditions give the perimeter of a square and require calculation of its area. The formula for the area of a square through the perimeter is derived from the value of the perimeter. Perimeter is the sum of the lengths of all sides of the figure. Because squared 4 equal sides, then it will be equal. From here we find the side of the figure. The area of a square according to the usual formula is calculated as follows: .
Let's look at an example of calculating the area of a square using its perimeter.