What is the perimeter of the square? Perimeter of a square and rectangle. Methods of definition and examples of solutions

Sometimes a person is faced with the immediate need to find the perimeter of a square. For example, you need to make a fence around a square area, wallpaper a square room, or decorate the walls of a square dance hall with mirrors. To calculate the amount of material needed, you need to make special calculations. And here, without knowing, you will have to purchase the material “by eye.” It’s okay if it’s inexpensive wallpaper, but where will the extra mirrors go? And even if there is a shortage of material, it is then quite difficult to find additional material of the same quality.

So, how do you find out what the perimeter of a square is? We know that a square has all sides equal. And if the perimeter is the sum of all sides of the polygon, then the perimeter of the square can be written as (q+q+q+q), where q is a value indicating the length of one side of the square. Naturally, it is most convenient to use multiplication here. So, the perimeter of a square is quadruple the length of its side or 4q, where q is the side.

But if only the area of ​​the square is known, the perimeter of which needs to be found out - what to do in this case? And everything is very simple here! From the known figure by which it is expressed, it is necessary to extract. Thus, the size of the side of the square will be found. Now you need to find the perimeter of the square using the formula derived above.

Another question is if you need to find the perimeter of a square along its diagonal. Here we should remember the Pythagorean theorem. Consider a square WERT with diagonal WR. WR divided the square into two right isosceles triangles. If the length of the diagonal is known (let us conventionally take it as z, and the side as u), then the size of the side of the square must be sought based on the formula: the square z is equal to twice the square u, from which we conclude: u is equal to the square root taken from half the square of the hypotenuse . Then we increase the result obtained by 4 times - here you have the perimeter of the square!

You can find the side of a square by the radius of the circle inscribed in it. After all, the inscribed circle touches all sides of the square, from which the conclusion is drawn that the diameter of the circle is equal to the length of the side of the square. And the diameter - everyone knows this - is twice the radius.

If the radius or circumscribed around the square is known, then here we see that all 4 vertices of the square are located on the circle. This means that the diameter of the circumscribed circle is equal to the length of the diagonal of the square. Taking this position as a given, you should then calculate the perimeter using the formula for finding the perimeter along its diagonal, discussed above.

Sometimes a problem is proposed in which you need to find out what is the perimeter of a square that is inscribed in an isosceles square in such a way that one corner of the square coincides with the right angle of the triangle. The leg of this geometric figure is known. Let us denote the triangle as WER, where vertex E is common.

The inscribed square will be labeled ETYU. The ET side lies on the WE side, and the EU side lies on the ER side. The vertex Y lies on the hypotenuse WR. Looking further at the drawing, we can draw the following conclusions:

1. WTY is an isosceles triangle, since according to the condition WER is isosceles, which means that the angle EWR is equal to 45 degrees, and the resulting triangle is right-angled with a base angle of also 45 degrees, which allows us to assert that it is isosceles. It follows that WT=TY.
2. TY=ET as the sides of a square.
3. Following the same algorithm, we derive the following: YU=UR, and UR=EU.
4. The sides of a triangle can be represented as the sum of segments. EW=ET+TW, and ER= EU+UR.
5. Replacing equal segments, we derive: EW=ET+TY, and ER=EU+UY.
6. If the perimeter of an inscribed square is expressed by the formula (ET+TY)+(EU+UY), then this can be written differently, taking into account the just derived values ​​of the sides of the triangle, as EW+ER. That is, the perimeter of a square inscribed in a right triangle with a coinciding right angle will be equal to the sum of its legs.

These, of course, are not all options for calculating the perimeter of a square, but only the most common ones. But they are all based on the fact that the perimeter of a quadrilateral is the sum of all its sides. And there is no escape from this!

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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.

Example.

It reads like this: quadrilateral ABCD; square EFGH.

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

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)

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

Formula for calculating the perimeter of a square

P ABCD = 2 * (AB + BC)

P ABCD = 4 * AB

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

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?

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 in which you live. Determine the perimeter of your room.

What is the area of ​​a rectangle?

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

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

Answer: 14 cm 2.

Formula for calculating the area of ​​a square

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

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

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.

To calculate the area and perimeter of a square, you need to understand the concepts of these quantities. A square is a rectangle with only four equal sides that have an angle of 90° to each other. The perimeter is the sum of the lengths of all sides. Area is the product of the length of a rectangular figure and its width.

Area of ​​a square and how to find it

As mentioned above, a square is a rectangle with 4 equal sides, so the answer to the question: “how to find the area of ​​a square” is the formula: S = a*a or S = a 2 , where a is the side of the square. Based on this formula, it is easy to find the side of a square if the area is known. To do this, you need to extract the square from the indicated value.

For example, S = 121, therefore, a = √121 = 11. If the given value is not in the table of squares, then you can use the calculator: S = 94, a = √94 = 9.7.

How to find the perimeter of a square

The perimeter of a square is found using the easy formula: P = 4a, where a is the side of the square.

Example:

• side of square = 5, therefore P = 4*5 = 20
• side of the square = 3, therefore P = 4*3 = 12

But there are problems where the area is clearly indicated, but you need to find the perimeter. When solving, you need the formulas that were presented earlier.

For example: how to find the perimeter of a square if the area is known to be 144?

Solution steps:

1. Find out the length of one side: a = √144 = 12
2. Find the perimeter: P = 4*12 = 48.

Finding the perimeter of an inscribed square

There are several other ways to find the perimeter of a square. Let's consider one of them: finding the perimeter through the radius of the circumscribed circle. Here a new term “inscribed square” appears - this is a square whose vertices lie on a circle.

Solution algorithm:

• since we are considering a square, the formula can be expressed as follows: a 2 + a 2 = (2r) 2 ;
• then the equation should be made simpler: 2a 2 = 4(r) 2 ;
• divide the equation by 2: (a 2 ) = 2(r) 2 ;
• extract the root: a = √(2r).

As a result, we get the last formula: a (side of the square) = √(2r).

1. The found side of the square is multiplied by 4, then the standard formula for finding the perimeter is applied: P = 4√(2r).

Task:

Given a square that is inscribed in a circle, its radius is 5. This means that the diagonal of the square is 10. We apply the Pythagorean theorem: 2(a 2 ) = 10 2 , that is 2a 2 = 100. Divide the result by two and the result is: a 2 = 50. Since this is not a tabular value, we use a calculator: a = √50 = 7.07. Multiply by 4: P = 4*7.07 = 28.2. Problem solved!

Let's consider one more question

Often in problems we encounter another condition: how to find the area of ​​a square if the perimeter is known?

We have already considered all the necessary formulas, so to solve problems of this type, it is necessary to skillfully apply them and connect them with each other. Let's move straight to an illustrative example: The area of ​​a square is 25 cm 2 , find its perimeter.

Solution steps:

1. Find the side of the square: a = √25 = 5.

1. We find the perimeter itself: P = 4*a = 4*5 = 20.

To summarize, it is important to recall that such simple formulas are applicable not only in educational activities, but also in everyday life. Children learn to find the perimeter and area of ​​a figure in elementary school. In the middle grades, a new subject appears - geometry, where the Pythagorean theorem is at the very beginning of study. These basics of mathematics are also tested at the end of the OGE and USE school, so it is important to know these formulas and apply them correctly.

Square(from lat. quadratus- quadrangular) - a regular quadrilateral in which all sides and angles are equal to each other. It can be defined as a rectangle in which two adjacent sides are equal to each other, or as a rhombus in which all angles are right.

Symmetry. The square has the greatest symmetry among all quadrilaterals. It has:

In this case, indicate the side of the square a, then the length of the diagonal d calculated using the Pythagorean axiom:

d = √(a2 +a2) = √(2a2) = √2·a.

Inscribed and circumscribed circles. A circle inscribed in a square touches the midpoint of all sides of the square and has a radius r, equal to half the side of the square a. A circle circumscribed around a square passes through all its vertices and has a radius R, equal to half the length of the diagonal of the square d:

r = a/2,

R = d/2 = (√2/2) a.

Perimeter and area. Perimeter P A square is made up of the lengths of its 4 sides. Square S square is equal to the square of its side length:

P = 4a = 8r = 2√2·R,

S = a2 = 4r2 = 2R2.

Sources:

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Often on the Internet you can find ridicule about how knowledge in mathematics - integrals, differentials, trigonometric functions and other sections of the subject - does not help make a person’s life easier. Such jokes are in vain, because how helpful is the ability to correctly calculate the perimeter of a square, rectangle and other geometric shapes in construction work. Material consumption: tiles, wallpaper, flooring cannot be determined without understanding basic mathematical formulas and geometric figures.

Properties of a square

Any calculations in mathematics are based on the properties of an object. To answer the question: “What is the perimeter of the square?” - It is recommended to remember the distinctive characteristics of this figure.

1. Equality on all sides.
2. Having four 90 degree angles.
3. Parallelism of sides.
4. Rotational symmetry. When you rotate a figure, its appearance remains unchanged.
5. Ability to describe and inscribe a circle.
6. When diagonals intersect, they bisect each other.
7. The area of ​​a figure characterizes the space filled by a square in two-dimensional space.
8. The perimeter of a figure is nothing more than the sum of the lengths of its sides.
9. From the previous property it follows that the units of measurement of the perimeter will be length units: m, cm, dm and others.

To count skirting boards to complete a renovation in a square room, you need to know the length of the room. To do this, you need to calculate its perimeter.

Perimeter

Translated from Greek, the word means “to measure around.” The term applies to all closed figures: square, circle, rectangle, triangle, trapezoid and others. Knowledge of determining the perimeter of elementary figures is necessary for solving complex geometric problems with irregularly shaped objects. For example, to calculate the baseboards for a room with a “G” type layout, or as it is also called a “boot”, you will need to determine the perimeter of a square and a rectangle. After all, the shape of the room consists of these elementary figures.

The generally accepted designation for such a value is the letter P. Each figure, taking into account its properties, has its own formula for determining the perimeter.

Rectangle Properties

1. Equality of opposite sides.
2. Equality of diagonals.
3. Ability to describe a circle.
4. The heights of a rectangle are equal to its sides.
5. The sum of the angles is 360 degrees, and all angles are right angles.
6. Parallelism of opposite sides.
7. Perpendicularity of adjacent sides.
8. The sum of the squares of the diagonals of a rectangle is equal to the sum of the squares of its sides.
9. Intersecting, the diagonals bisect each other.
10. Inability to fit a circle into a figure.

Perimeter of a square

Depending on the established (known) parameters of the square, there are different formulas for determining its perimeter. A simple task is to calculate the perimeter given the length of its side (c). In this case, P=c+c+c+c or 4*c. For example, the side length of a square is 7 cm, then the perimeter of the figure will be 28 cm (4*7).

In the first case, everything is clear, but how to find the perimeter of a square, knowing its area? And here everything is extremely clear. Since the area of ​​a figure is determined by multiplying one side by the other, and a square has all sides equal, it is necessary to take the root of a known quantity. Example: there is a square with an area of ​​25 dm 2. The root of 25 is equal to 5 - this value characterizes the length of the side of the square. Now, by substituting the found value - 5 dm 2 - into the original perimeter formula, we can solve the problem. The answer will be a value of 20 dm. That is, 4 multiplied by 5, we got the desired value.

Square and circle

From the properties of the figure in question, it emerges that a circle can be inscribed in a square and also described around the figure.

The first option is to find the perimeter along the radius of the circumscribed circle. A square whose vertices are on a circle is considered inscribed. The radius of the circle is equal to 1/2 the length of the diagonal. It turns out that the diameter is equal to the diagonal. Now we need to consider the right triangle, which is the result of dividing a square with a diagonal. Solving the problem comes down to finding the sides of this triangle. BC is a known quantity, the diameter of a circumscribed circle. Let's say it is equal to 3 cm. The Pythagorean theorem in the case of equal sides of a triangle will look like this: 2c 2 = 3 2. In the formula, the notation c is the length of the side of the triangle and square; 3 is the known value of the hypotenuse. Hence, c=√9/2. Knowing the side of a square, calculating its perimeter is not a problem.

The peculiarity of the inscribed circle is that the sides of the square are divided in half. Therefore, the radius is equal to half the length of the side of the square. Then side c=2*radius. The perimeter of the square in this case is equal to 4 * 2 * radius or 8 radii of the circle.

Perimeter of a rectangle

The most elementary formula for determining the perimeter of a rectangle through the known values ​​of its sides looks like this: P = 2 (a + b), where a and b are the lengths of the sides of the figure.

The diagonal of a rectangle, similar to a square, divides the figure in half, forming a right triangle. However, the task is complicated by the fact that the sides of this triangle are unequal. In the case of a known size of one of the sides and the diagonal, the second can be found by following the Pythagorean theorem: d 2 = a 2 + b 2, where a and b are the sides of the figure, and d is the diagonal.

If neither side is known, then knowledge of trigonometry comes into play: sines, cosines and other functions.

Finding the perimeter of a circumscribed circle and a known diameter comes down to the fact that the diameter is equal to the length of the diagonal of the figure. Further, the solution to the problem is determined by the presence of known quantities. If angles are given, then through trigonometric functions. If a side is given, the answer will be found through the Pythagorean theorem.

Rectangle and trigonometric functions

For clarity, an example of solving the problem is given. Given: rectangle ABCD; diagonal length ( d) 20 cm; corner f- 30°. Find the perimeter of the figure.

From trigonometry, you need to remember the following: the sine of an angle in a right triangle is equal to the ratio of the opposite side to the hypotenuse. The sine of 30° (there are tables from which you can determine the values ​​of trigonometric functions for regular angles) is equal to 1/2. It turns out 1/2 = ratio in to d. The unknown quantity in will be equal to d/2=20/2=10 cm.

To calculate the perimeter, you need to find the second side of the figure. It is possible through the Pythagorean theorem, since the lengths of the hypotenuse and one of the legs are known, or again through the ratio of the sides for the cosine of the angle.

Cosine of angle f expressed as the ratio of the adjacent leg to the hypotenuse and is equal to √3/2.

√3/2=n/d, n=(d*√3)/2 or 10*√3. After taking the root of 3, we get the length of the side of the triangle: 10 * 1.73 = 17.3 cm.

The perimeter is 2(17.3+10)=2*27.3=54.6 cm.

Perimeter and aspect ratio

In the school curriculum there are geometry problems where the lengths of the sides of a rectangle are expressed by their ratio to each other. A discussion of the solution to such a problem is presented below.

It is known that the sum of the lengths of all sides of a rectangle, that is, its perimeter, is 84 cm. The ratio of length (l) to width (w) is 3:2. Find the sides of the figure.

Solution: let the length be 3x and the width 2x, according to the ratio from the problem statement. The formula for the perimeter of a rectangle with the obtained side lengths will be as follows: 3x + 3x + 2x + 2x = 84. Next, 10x = 84, x = 8.4 cm. By substituting x into the expression for the length and width of the rectangle, you can find the required values. The length will be: 3*8.4 = 25.2 cm; width: 2*8.4 = 16.8 cm.

The article is devoted to solving the most common problems in the school curriculum. And these are not all the ways to find the perimeter of a square and rectangle.