Area square We calculate the area of a square: along the side, diagonal, perimeter

Area of a polygon

We will associate the concept of area of a polygon with such a geometric figure as 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 $3$.

Square is a regular quadrilateral in which all sides and angles are equal to each other.
The area of a square is equal to the square of its side:
S = a 2

Proof

Let's start with the case when a = 1/n, where n is an integer.
Let's take a square with side 1 and divide it into n 2 equal squares as shown in Figure 1.

Since the area of the large square is equal to one, the area of each small square is equal to 1/n 2. The side of each small square is 1/n, i.e. equal to a. So,
S = 1/n 2 = (1/n) 2 = a 2 . (1)
Let now the number a represents a finite decimal fraction containing n decimal places (in particular, the number a can be an integer, in which case n = 0). Then the number m = a · 10 n is an integer. Let us divide this square with side a into m 2 equal squares as shown in Figure 2.

Moreover, each side of a given square will be divided into m equal parts, and, therefore, the side of any small square is equal to

a/m = a / (a · 10 n) = 1/10 n.

According to the formula (1) The area of the small square is (1/10 n) 2 . Hence, The area S of this square is equal to

m 2 · (1/10 n) 2 = (m/10 n) 2 = ((a · 10 n)/10 n) 2 = a 2 .

Finally, let the number a represents an infinite decimal fraction. Consider the number a n, obtained from a by discarding all decimal places starting from (n+1) th. Since the number a differs from a n no more than 1/10 n, That a n ≤ a ≤ a n + 1/10 n, where

a n 2 ≤ a 2 ≤ (a n + 1/10 n) 2 . (2)

It is clear that the area S of a given square is enclosed between the area of a square with side a n and the area of a square with side a n + 1/10 n:

i.e. between a n 2 And (a n + 1/10 n) 2:

a n 2 ≤ S ≤ (a n + 1/10 n) 2 . (3)

We will increase the number unlimitedly n. Then the number 1/10 n will become arbitrarily small, and, therefore, the number (a n + 1/10 n) 2 will differ as little as desired from the number a n 2. Therefore, from the inequalities (2) And (3) it follows that the number S differs as little as desired from the number a 2 . Therefore, these numbers are equal: S = a 2, which was what needed to be proven.

The area of a square can also be found using the following formulas:

S = 4r 2 ,
S = 2R 2,

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 also 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.

Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour 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 is 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,

Additional materials
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Teaching aids and simulators in the Integral online store for grade 3
Trainer for 3rd grade "Rules and exercises in mathematics"
Electronic textbook for grade 3 "Math in 10 minutes"

What are rectangle and square

Rectangle is a quadrilateral with all right angles. This means that opposite sides are equal to each other.

Square is a rectangle with equal sides and equal angles. It is called a regular quadrilateral.

Quadrangles, including rectangles and squares, are designated by 4 letters - vertices. Latin letters are used to designate vertices: A, B, C, D...

Example.

It reads like this: quadrilateral ABCD; square EFGH.

What is the perimeter of a rectangle? Formula for calculating perimeter

Perimeter of a rectangle is the sum of the lengths of all sides of the rectangle or the sum of the length and width multiplied by 2.

The perimeter is indicated by a Latin letter P. Since the perimeter is the length of all sides of the rectangle, the perimeter is written in units of length: mm, cm, m, dm, km.

For example, the perimeter of rectangle ABCD is denoted as P ABCD, where A, B, C, D are the vertices of the rectangle.

Let's write down the formula for the perimeter of a quadrilateral ABCD:

P ABCD = AB + BC + CD + AD = 2 * AB + 2 * BC = 2 * (AB + BC)

Example.
Given a rectangle ABCD with sides: AB=CD=5 cm and AD=BC=3 cm.
Let's define P ABCD.

Solution:
1. Let's draw a rectangle ABCD with the original data.
2. Let’s write a formula to calculate the perimeter of a given rectangle:

P ABCD = 2 * (AB + BC)

P ABCD = 2 * (5 cm + 3 cm) = 2 * 8 cm = 16 cm

Answer: P ABCD = 16 cm.

Formula for calculating the perimeter of a square

We have a formula for determining the perimeter of a rectangle.

P ABCD = 2 * (AB + BC)

Let's use it to determine the perimeter of a square. Considering that all sides of the square are equal, we get:

P ABCD = 4 * AB

Example.
Given a square ABCD with a side equal to 6 cm. Let us determine the perimeter of the square.

Solution.
1. Let's draw a square ABCD with the original data.

2. Let us recall the formula for calculating the perimeter of a square:

P ABCD = 4 * AB

3. Let’s substitute our data into the formula:

P ABCD = 4 * 6 cm = 24 cm

Answer: P ABCD = 24 cm.

Problems to find the perimeter of a rectangle

1. Measure the width and length of the rectangles. Determine their perimeter.

2. Draw a rectangle ABCD with sides 4 cm and 6 cm. Determine the perimeter of the rectangle.

3. Draw a square SEOM with a side of 5 cm. Determine the perimeter of the square.

Where is the calculation of the perimeter of a rectangle used?

1. A plot of land has been given; it needs to be surrounded by a fence. How long will the fence be?

In this task, it is necessary to accurately calculate the perimeter of the site so as not to buy excess material for building a fence.

2. Parents decided to renovate the children's room. You need to know the perimeter of the room and its area in order to correctly calculate the amount of wallpaper.
Determine the length and width of the room you live in. Determine the perimeter of your room.

What is the area of a rectangle?

Square is a numerical characteristic of a figure. Area is measured in square units of length: cm 2, m 2, dm 2, etc. (centimeter squared, meter squared, decimeter squared, etc.)
In calculations it is denoted by a Latin letter S.

To determine the area of a rectangle, multiply the length of the rectangle by its width.
The area of the rectangle is calculated by multiplying the length of the AC by the width of the CM. Let's write this down as a formula.

S AKMO = AK * KM

Example.
What is the area of rectangle AKMO if its sides are 7 cm and 2 cm?

S AKMO = AK * KM = 7 cm * 2 cm = 14 cm 2.

Answer: 14 cm 2.

Formula for calculating the area of a square

The area of a square can be determined by multiplying the side by itself.

Example.
In this example, the area of a square is calculated by multiplying the side AB by the width BC, but since they are equal, the result is multiplying the side AB by AB.

S ABCO = AB * BC = AB * AB

Example.
Determine the area of a square AKMO with a side of 8 cm.

S AKMO = AK * KM = 8 cm * 8 cm = 64 cm 2

Answer: 64 cm 2.

Problems to find the area of a rectangle and square

1. Given a rectangle with sides 20 mm and 60 mm. Calculate its area. Write your answer in square centimeters.

2. A dacha plot measuring 20 m by 30 m was purchased. Determine the area of the dacha plot and write the answer in square centimeters.