Knowledge of how to measure the Earth appeared in ancient times and gradually took shape in the science of geometry. WITH Greek language This word is translated as “land surveying”.
The measure of the extent of a flat section of the Earth in length and width is area. In mathematics, it is usually denoted by the Latin letter S (from the English “square” - “area”, “square”) or the Greek letter σ (sigma). S denotes the area of a figure on a plane or the surface area of a body, and σ is the area cross section wires in physics. These are the main symbols, although there may be others, for example, in the field of strength of materials, A is the cross-sectional area of the profile.
Knowing the area simple figures, you can find parameters of more complex. Ancient mathematicians developed formulas that can be used to easily calculate them. Such figures are triangle, quadrangle, polygon, circle.
To find the area of a complex plane figure, it is broken down into many simple figures such as triangles, trapezoids or rectangles. Then, using mathematical methods, a formula is derived for the area of this figure. A similar method is used not only in geometry, but also in mathematical analysis to calculate the areas of figures bounded by curves.
Let's start with the simplest figure - a triangle. They are rectangular, isosceles and equilateral. Take any triangle ABC with sides AB=a, BC=b and AC=c (∆ ABC). To find its area, let us recall the sine and cosine theorems known from the school mathematics course. Letting go of all calculations, we arrive at the following formulas:
Let there be a quadrilateral ABCD with AB=a, BC=b, CD=c, AD=d. To find the area S of an arbitrary 4-gon, you need to divide it by the diagonal into two triangles, the areas of which S1 and S2 are not equal in the general case.
Then use the formulas to calculate them and add them, i.e. S=S1+S2. However, if a 4-gon belongs to a certain class, then its area can be found using previously known formulas:
To find the area of an n-gon, mathematicians break it down into the simplest equal figures - triangles, find the area of each of them and then add them. But if the polygon belongs to the class of regular, then use the formula:
S=a n h/2=a² n/=P²/, where n is the number of vertices (or sides) of the polygon, a is the side of the n-gon, P is its perimeter, h is the apothem, i.e. a segment drawn from the center of the polygon to one of its sides at an angle of 90°.
A circle is a perfect polygon with an infinite number of sides. We need to calculate the limit of the expression on the right in the formula for the area of a polygon with the number of sides n tending to infinity. In this case, the perimeter of the polygon will turn into the length of a circle of radius R, which will be the boundary of our circle, and will become equal to P=2 π R. Substitute this expression into the above formula. We will get:
S=(π² R² cos (180°/n))/(n sin (180°/n)).
Let's find the limit of this expression as n→∞. To do this, we take into account that lim (cos (180°/n)) for n→∞ is equal to cos 0°=1 (lim is the sign of the limit), and lim = lim for n→∞ is equal to 1/π (we converted the degree measure into a radian, using the relation π rad=180°, and applied the first remarkable limit lim (sin x)/x=1 at x→∞). Substituting in last expression for S the obtained values, we arrive at well-known formula:
S=π² R² 1 (1/π)=π R².
Systemic and non-systemic units of measurement are used. System units belong to the SI (System International). This is a square meter (sq. meter, m²) and units derived from it: mm², cm², km².
IN square millimeters(mm²), for example, measure the cross-sectional area of wires in electrical engineering, in square centimeters (cm²) - the cross-section of a beam in structural mechanics, in square meters(m²) - apartments or houses, in square kilometers(km²) - territories in geography.
However, sometimes non-systemic units of measurement are used, such as: weave, ar (a), hectare (ha) and acre (as). Let us present the following relations:
A rectangle is a special case of a quadrilateral. This means that the rectangle has four sides. Its opposite sides are equal: for example, if one of its sides is 10 cm, then the opposite side will also be equal to 10 cm. A special case of a rectangle is a square. A square is a rectangle with all sides equal. To calculate the area of a square, you can use the same algorithm as to calculate the area of a rectangle.
In order to find the area of a rectangle, you need to multiply its length by its width: Area = Length × Width. In the case given below: Area = AB × BC.
Some problems require you to find the area of a rectangle using the length of the diagonal and one of the sides. The diagonal of a rectangle divides it into two equal right triangles. Therefore, we can determine the second side of the rectangle using the Pythagorean theorem. After this, the task is reduced to the previous point.
The perimeter of a rectangle is the sum of all its sides. If you know the perimeter of the rectangle and one side (such as the width), you can calculate the area of the rectangle using the following formula:
Area = (Perimeter×width – width^2)/2.
The diagonals in a rectangle are equal, so to calculate the area based on the length of the diagonal and the sine acute angle between them, you should use the following formula: Area = Diagonal^2 × sin(acute angle between diagonals)/2.
The area of a rectangle may not sound arrogant, but it is an important concept. IN Everyday life we are constantly faced with it. Find out the size of fields, vegetable gardens, calculate the amount of paint needed to whitewash the ceiling, how much wallpaper will be needed for pasting
money and more.
Geometric figure
First, let's talk about the rectangle. This is a figure on a plane that has four right angles and its opposite sides are equal. Its sides are usually called length and width. They are measured in millimeters, centimeters, decimeters, meters, etc. Now we will answer the question: “How to find the area of a rectangle?” To do this, you need to multiply the length by the width.
Area=length*width
But one more caveat: length and width must be expressed in the same units of measurement, that is, meter and meter, and not meter and centimeter. The area is written with the Latin letter S. For convenience, let’s denote the length with the Latin letter b, and the width with the Latin letter a, as shown in the figure. From this we conclude that the unit of area is mm 2, cm 2, m 2, etc.
Let's look at specific example How to find the area of a rectangle. Length b=10 units. Width a=6 units. Solution: S=a*b, S=10 units*6 units, S=60 units 2. Task. How to find out the area of a rectangle if the length is 2 times the width and is 18 m? Solution: if b=18 m, then a=b/2, a=9 m. How to find the area of a rectangle if both sides are known? That's right, substitute it into the formula. S=a*b, S=18*9, S=162 m 2. Answer: 162 m2. Task. How many rolls of wallpaper do you need to buy for a room if its dimensions are: length 5.5 m, width 3.5, and height 3 m? Dimensions of a roll of wallpaper: length 10 m, width 50 cm. Solution: make a drawing of the room.
The areas of opposite sides are equal. Let's calculate the area of a wall with dimensions of 5.5 m and 3 m. S wall 1 = 5.5 * 3,
S wall 1 = 16.5 m 2. Therefore, the opposite wall has an area of 16.5 m2. Let's find the area of the next two walls. Their sides, respectively, are 3.5 m and 3 m. S wall 2 = 3.5 * 3, S wall 2 = 10.5 m 2. This means that the opposite side is also equal to 10.5 m2. Let's add up all the results. 16.5+16.5+10.5+10.5=54 m2. How to calculate the area of a rectangle if the sides are expressed in different units of measurement. Previously, we calculated areas in m2, and in this case we will use meters. Then the width of the wallpaper roll will be equal to 0.5 m. S roll = 10 * 0.5, S roll = 5 m 2. Now we’ll find out how many rolls are needed to cover a room. 54:5=10.8 (rolls). Since they are measured in whole numbers, you need to buy 11 rolls of wallpaper. Answer: 11 rolls of wallpaper. Task. How to calculate the area of a rectangle if it is known that the width is 3 cm shorter than the length, and the sum of the sides of the rectangle is 14 cm? Solution: let the length be x cm, then the width is (x-3) cm. x+(x-3)+x+(x-3)=14, 4x-6=14, 4x=20, x=5 cm - length rectangle, 5-3=2 cm - width of the rectangle, S=5*2, S=10 cm 2 Answer: 10 cm 2.
Summary
Having looked at the examples, I hope it has become clear how to find the area of a rectangle. Let me remind you that the units of measurement for length and width must match, otherwise you will get an incorrect result. To avoid mistakes, read the task carefully. Sometimes a side can be expressed through the other side, don't be afraid. Please refer to our solved problems, it is quite possible that they can help. But at least once in our lives we are faced with finding the area of a rectangle.
We have to deal with such a concept as area in our daily lives. So, for example, when building a house you need to know it in order to calculate the amount required material. Size garden plot will also be characterized by area. Even renovations in an apartment cannot be done without this definition. Therefore, the question of how to find the area of a rectangle comes up very often and is important not only for schoolchildren.
For those who don't know, a rectangle is a flat figure in which opposite sides are equal and the angles are 90 degrees. To denote area in mathematics we use English letter S. It is measured in square units: meters, centimeters, and so on.
Now we will try to give a detailed answer to the question of how to find the area of a rectangle. There are several ways to determine this value. Most often we come across a method of determining area using width and length.
Let's take a rectangle with width b and length k. To calculate the area of a given rectangle, you need to multiply the width by the length. All this can be represented in the form of a formula that will look like this: S = b * k.
Now let's look at this method using a specific example. It is necessary to determine the area of a garden plot with a width of 2 meters and a length of 7 meters.
S = 2 * 7 = 14 m2
In mathematics, especially in mathematics, we have to determine the area in other ways, since in many cases we do not know either the length or width of the rectangle. At the same time, other known quantities exist. How to find the area of a rectangle in this case?
In formula form it will look like this:
S = cos(a) * sin(a) * d2, where d is the length of the diagonal
It happens that instead of the radius, we know the diameter of the inscribed circle. Then the formula will look like this:
S=d2, where d is the diameter.
S=b* (P - 2*b), where b is the length of the side, P is the perimeter.
As you can see, the area of a rectangle can be determined different ways. It all depends on what quantities we know before considering this issue. Of course, the latest calculus methods are practically never encountered in life, but they can be useful for solving many problems in school. Perhaps this article will be useful for solving your problems.
Instructions
To find length sides of the rectangle, if known width And square, divide the area number by the width number. That is, use the formula: L = P / W, where: D is the length of the side of the rectangle,
Ш – width rectangle,
P - his square.For example, if square rectangle is 20 cm², and its width– 5 cm, then the length of its side will be: 20 / 5 = 4 cm.
Before starting calculations, translate the width and square rectangle into one measurement system. That is, square must be expressed in square units corresponding to the width. In this case, the length will be in the same units as width. So, if width is given in meters, then square necessary in . This is especially relevant when measuring land plots, where square usually given in hectares, acreages and “hundreds”.
For example, let square summer cottage equals six hundred square meters, and his width– 30 meters. Need to find length plot.
Since “hundredth” is 100, then square a “standard” six can be written as 600 m². From here length land plot can be found by dividing 600 by 30. It turns out - 20 meters.
Sometimes given square And width a figure that has not a rectangular, but an arbitrary shape. At the same time, you also need to find it length. As a rule, in this case, dimensional figures are meant, that is, the parameters of the rectangle in which this figure can be enclosed.
If greater calculation accuracy is not required, then use the above formula (L = P / W). However, the length value will be underestimated. To get more exact value length of the figure, estimate how completely the figure fills its overall rectangle and divide the resulting length on the fill factor.
Sources:
Each geometric figure has certain characteristics, which, in turn, are interconnected. Therefore, in order to find the area of a rectangle, you need to know the length of its sides.
The rectangle is one of the most common geometric shapes. It is a quadrilateral, all angles of which are equal to each other and amount to 90 degrees. This characteristic, in turn, entails certain consequences in relation to other parameters of the figure in question.
Firstly, its sides, located opposite each other, will be parallel. Secondly, these sides will be pairwise equal in length. These characteristics turn out to be very important for calculating its other parameters, such as area.
Therefore, to designate the different sides of a rectangle, special notations are adopted: for example, the side with a large extent is usually called the length of the figure, and the side with a smaller extent is its width. Moreover, each rectangle, due to its properties described above, has two lengths and two widths.
The actual algorithm for calculating the area of this figure is quite simple: you only need to multiply its one length by one of its width. The resulting product will be the area of the rectangle.
To find the area of a figure, you need to multiply its width by its length: thus, the area of the rectangle in question will be 40 square centimeters. Please note that for calculations to take place, both parameters used must be measured in the same units, for example
