Help on Tele2, tariffs, questions

In order to feel confident and successfully solve problems in geometry lessons, it is not enough to learn the formulas. They need to be understood first. To be afraid, and even more so to hate formulas, is unproductive. In this article accessible language Various methods for finding the area of ​​a trapezoid will be analyzed. To better understand the corresponding rules and theorems, we will pay some attention to its properties. This will help you understand how the rules work and in what cases certain formulas should be applied.

Defining a trapezoid

What kind of figure is this overall? A trapezoid is a polygon with four corners and two parallel sides. The other two sides of the trapezoid can be inclined at different angles. Its parallel sides are called bases, and for non-parallel sides the name “sides” or “hips” is used. Such figures are quite common in everyday life. The contours of the trapezoid can be seen in the silhouettes of clothing, interior items, furniture, dishes and many others. Trapeze happens different types: scalene, equilateral and rectangular. We will examine their types and properties in more detail later in the article.

Properties of a trapezoid

Let us dwell briefly on the properties of this figure. The sum of the angles adjacent to any side is always 180°. It should be noted that all angles of a trapezoid add up to 360°. The trapezoid has the concept of a midline. If you connect the midpoints of the sides with a segment, this will be the middle line. It is designated m. The middle line has important properties: it is always parallel to the bases (we remember that the bases are also parallel to each other) and equal to their half-sum:

This definition must be learned and understood, because it is the key to solving many problems!

With a trapezoid, you can always lower the height to the base. An altitude is a perpendicular, often denoted by the symbol h, that is drawn from any point of one base to another base or its extension. The midline and height will help you find the area of ​​the trapezoid. Such tasks are the most common in school course geometry and regularly appear among test and examination papers.

The simplest formulas for the area of ​​a trapezoid

Let's look at the two most popular and simple formulas used to find the area of ​​a trapezoid. It is enough to multiply the height by half the sum of the bases to easily find what you are looking for:

S = h*(a + b)/2.

In this formula, a, b denote the bases of the trapezoid, h - the height. For ease of perception, in this article, multiplication signs are marked with a symbol (*) in formulas, although in official reference books the multiplication sign is usually omitted.

Let's look at an example.

Given: a trapezoid with two bases equal to 10 and 14 cm, the height is 7 cm. What is the area of ​​the trapezoid?

Let's look at the solution to this problem. Using this formula, you first need to find the half-sum of the bases: (10+14)/2 = 12. So, the half-sum is equal to 12 cm. Now we multiply the half-sum by the height: 12*7 = 84. What we are looking for is found. Answer: The area of ​​the trapezoid is 84 square meters. cm.

The second well-known formula says: the area of ​​a trapezoid is equal to the product of the midline and the height of the trapezoid. That is, it actually follows from the previous concept of the middle line: S=m*h.

Using diagonals for calculations

Another way to find the area of ​​a trapezoid is actually not that complicated. It is connected to its diagonals. Using this formula, to find the area, you need to multiply the half-product of its diagonals (d 1 d 2) by the sine of the angle between them:

S = ½ d 1 d 2 sin a.

Let's consider a problem that shows the application of this method. Given: a trapezoid with the length of the diagonals equal to 8 and 13 cm, respectively. The angle a between the diagonals is 30°. Find the area of ​​the trapezoid.

Solution. Using the above formula, it is easy to calculate what is required. As you know, sin 30° is 0.5. Therefore, S = 8*13*0.5=52. Answer: the area is 52 square meters. cm.

Finding the area of ​​an isosceles trapezoid

A trapezoid can be isosceles (isosceles). Its sides are the same and the angles at the bases are equal, which is well illustrated by the figure. An isosceles trapezoid has the same properties as a regular one, plus a number of special ones. A circle can be circumscribed around an isosceles trapezoid, and a circle can be inscribed within it.

What methods are there for calculating the area of ​​such a figure? The method below will require a lot of calculations. To use it, you need to know the values ​​of the sine (sin) and cosine (cos) of the angle at the base of the trapezoid. To calculate them, you need either Bradis tables or an engineering calculator. Here is the formula:

S= c*sin a*(a - c*cos a),

Where With- lateral thigh, a- angle at the lower base.

An equilateral trapezoid has diagonals of equal length. The converse is also true: if a trapezoid has equal diagonals, then it is isosceles. Hence the following formula to help find the area of ​​a trapezoid - the half product of the square of the diagonals and the sine of the angle between them: S = ½ d 2 sin a.

Finding the area of ​​a rectangular trapezoid

Famous special case rectangular trapezoid. This is a trapezoid, in which one side (its thigh) adjoins the bases at a right angle. It has the properties of a regular trapezoid. In addition, she has very interesting feature. The difference in the squares of the diagonals of such a trapezoid is equal to the difference in the squares of its bases. All previously described methods for calculating area are used for it.

We use ingenuity

There is one trick that can help if you forget specific formulas. Let's take a closer look at what a trapezoid is. If we mentally divide it into parts, we will get familiar and understandable geometric shapes: a square or rectangle and a triangle (one or two). If the height and sides of the trapezoid are known, you can use the formulas for the area of ​​a triangle and a rectangle, and then add up all the resulting values.

Let's illustrate this with the following example. Given a rectangular trapezoid. Angle C = 45°, angles A, D are 90°. The upper base of the trapezoid is 20 cm, the height is 16 cm. You need to calculate the area of ​​the figure.

This figure obviously consists of a rectangle (if two angles are equal to 90°) and a triangle. Since the trapezoid is rectangular, therefore, its height is equal to its side, that is, 16 cm. We have a rectangle with sides of 20 and 16 cm, respectively. Now consider a triangle whose angle is 45°. We know that one side of it is 16 cm. Since this side is also the height of the trapezoid (and we know that the height descends to the base at a right angle), therefore, the second angle of the triangle is 90°. Hence the remaining angle of the triangle is 45°. The consequence of this is that we get a right isosceles triangle with two equal sides. This means that the other side of the triangle is equal to the height, that is, 16 cm. It remains to calculate the area of ​​the triangle and the rectangle and add the resulting values.

The area of ​​a right triangle is equal to half the product of its legs: S = (16*16)/2 = 128. The area of ​​a rectangle is equal to the product of its width and length: S = 20*16 = 320. We found the required: area of ​​the trapezoid S = 128 + 320 = 448 sq. see. You can easily double-check yourself using the above formulas, the answer will be identical.

We use the Pick formula

Finally, we present another original formula that helps to find the area of ​​a trapezoid. It is called the Pick formula. It is convenient to use when the trapezoid is drawn on checkered paper. Similar problems are often found in GIA materials. It looks like this:

S = M/2 + N - 1,

in this formula M is the number of nodes, i.e. intersections of the lines of the figure with the lines of the cell at the boundaries of the trapezoid (orange dots in the figure), N is the number of nodes inside the figure (blue dots). It is most convenient to use it when finding the area of ​​an irregular polygon. However, the larger the arsenal of techniques used, the more less mistakes and better results.

Of course, the information provided does not exhaust the types and properties of a trapezoid, as well as methods for finding its area. This article provides an overview of its most important characteristics. When solving geometric problems, it is important to act gradually, start with easy formulas and problems, consistently consolidate your understanding, and move to another level of complexity.

Collected together the most common formulas will help students navigate the various ways to calculate the area of ​​a trapezoid and better prepare for tests and tests on this topic.

AND . Now we can begin to consider the question of how to find the area of ​​a trapezoid. This task arises very rarely in everyday life, but sometimes it turns out to be necessary, for example, to find the area of ​​a room in the shape of a trapezoid, which is increasingly used in construction modern apartments, or in renovation design projects.

A trapezoid is a geometric figure formed by four intersecting segments, two of which are parallel to each other and are called the bases of the trapezoid. The other two segments are called the sides of the trapezoid. In addition, we will need another definition later. This is the middle line of the trapezoid, which is a segment connecting the midpoints of the sides and the height of the trapezoid, which is equal to the distance between the bases.
Like triangles, trapezoids have special types in the form of an isosceles (equal-sided) trapezoid, in which the lengths of the sides are the same, and a rectangular trapezoid, in which one of the sides forms a right angle with the bases.

Trapezes have some interesting properties:

1. The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.
   
2. Isosceles trapezoids have equal sides and the angles they form with the bases.
   
3. The midpoints of the diagonals of a trapezoid and the point of intersection of its diagonals are on the same straight line.
   
4. If the sum of the sides of a trapezoid is equal to the sum of the bases, then a circle can be inscribed in it
   
5. If the sum of the angles formed by the sides of a trapezoid at any of its bases is 90, then the length of the segment connecting the midpoints of the bases is equal to their half-difference.
   
6. An isosceles trapezoid can be described by a circle. And vice versa. If a trapezoid fits into a circle, then it is isosceles.
   
7. The segment passing through the midpoints of the bases of an isosceles trapezoid will be perpendicular to its bases and represents the axis of symmetry.
   

How to find the area of ​​a trapezoid.

The area of ​​the trapezoid will be equal to half the sum of its bases multiplied by its height. In formula form, this is written as an expression:

where S is the area of ​​the trapezoid, a, b is the length of each of the bases of the trapezoid, h is the height of the trapezoid.

You can understand and remember this formula as follows. As follows from the figure below, using the center line, a trapezoid can be converted into a rectangle, the length of which will be equal to half the sum of the bases.

You can also expand any trapezoid into more simple figures: a rectangle and one or two triangles, and if it’s easier for you, then find the area of ​​the trapezoid as the sum of the areas of its constituent figures.

There is another simple formula for calculating its area. According to it, the area of ​​a trapezoid is equal to the product of its midline by the height of the trapezoid and is written in the form: S = m*h, where S is the area, m is the length of the midline, h is the height of the trapezoid. This formula more suitable for math problems than for everyday tasks, since in real conditions you will not know the length of the center line without preliminary calculations. And you will only know the lengths of the bases and sides.

In this case, the area of ​​the trapezoid can be found using the formula:

S = ((a+b)/2)*√c 2 -((b-a) 2 +c 2 -d 2 /2(b-a)) 2

where S is the area, a, b are the bases, c, d are the sides of the trapezoid.

There are several other ways to find the area of ​​a trapezoid. But, they are about as inconvenient as the last formula, which means there is no point in dwelling on them. Therefore, we recommend that you use the first formula from the article and wish you to always get accurate results.

Trapeze is called a quadrilateral whose only two the sides are parallel to each other.

They are called the bases of the figure, the rest are called the sides. Parallelograms are considered special cases of the figure. There is also a curved trapezoid, which includes the graph of a function. Formulas for the area of ​​a trapezoid include almost all of its elements, and The best decision is selected depending on the specified values.
The main roles in the trapezoid are assigned to the height and midline. middle line- This is a line connecting the midpoints of the sides. Height trapezoid is held at right angles from top corner to the base.
The area of ​​a trapezoid through its height is equal to the product of half the sum of the lengths of the bases multiplied by the height:

If the average line is known according to the conditions, then this formula is significantly simplified, since it is equal to half the sum of the lengths of the bases:

If, according to the conditions, the lengths of all sides are given, then we can consider an example of calculating the area of ​​a trapezoid using these data:

Suppose we are given a trapezoid with bases a = 3 cm, b = 7 cm and sides c = 5 cm, d = 4 cm. Let’s find the area of ​​the figure:

Area of ​​an isosceles trapezoid

An isosceles trapezoid, or, as it is also called, an isosceles trapezoid, is considered a separate case.
A special case is finding the area of ​​an isosceles (equilateral) trapezoid. The formula is derived different ways– through diagonals, through angles adjacent to the base and the radius of the inscribed circle.
If the length of the diagonals is specified according to the conditions and the angle between them is known, you can use the following formula:

Remember that the diagonals of an isosceles trapezoid are equal to each other!

That is, knowing one of their bases, side and angle, you can easily calculate the area.

Area of ​​a curved trapezoid

A special case is curved trapezoid. It is located on the coordinate axis and is limited by the graph of a continuous positive function.

Its base is located on the X axis and is limited to two points:
Integrals help calculate the area of ​​a curved trapezoid.
The formula is written like this:

Let's consider an example of calculating the area of ​​a curved trapezoid. The formula requires some knowledge to work with certain integrals. First, let's look at the value of the definite integral:

Here F(a) is the value of the antiderivative function f(x) at point a, F(b) is the value of the same function f(x) at point b.

Now let's solve the problem. The figure shows a curved trapezoid bounded by the function. Function
We need to find the area of ​​the selected figure, which is a curvilinear trapezoid bounded above by the graph, on the right by the straight line x =(-8), on the left by the straight line x =(-10) and the OX axis below.
We will calculate the area of ​​this figure using the formula:

The conditions of the problem give us a function. Using it we will find the values ​​of the antiderivative at each of our points:


Now
Answer: The area of ​​a given curved trapezoid is 4.

There is nothing complicated in calculating this value. The only thing that is important is extreme care in calculations.

What is an isosceles trapezoid? This is a geometric figure whose opposite, non-parallel sides are equal. There are several different formulas for finding the area of ​​a trapezoid with different conditions, which are given in the tasks. That is, the area can be found if the height, sides, angles, diagonals, etc. are given. It is also impossible not to mention that for isosceles trapezoids there are some “exceptions”, thanks to which the search for area and the formula itself are significantly simplified. Below are detailed solutions for each case with examples.

Necessary properties for finding the area of ​​an isosceles trapezoid

We have already found out that a geometric figure that has opposite, not parallel, but equal sides is a trapezoid, and an isosceles one. There are special cases when a trapezoid is considered isosceles.

• These are the conditions for equality of angles. So, mandatory item: the angles at the base (take the picture below) should be equal. In our case, angle BAD = angle CDA, and angle ABC = angle BCD
• Second important rule– in such a trapezoid the diagonals must be equal. Therefore, AC = BD.
• Third aspect: the opposite angles of the trapezoid must add up to 180 degrees. This means that angle ABC + angle CDA = 180 degrees. The same applies to angles BCD and BAD.
• Fourthly, if a trapezoid allows a circle to be described around it, then it is isosceles.

How to find the area of ​​an isosceles trapezoid - formulas and their descriptions

• S = (a+b)h/2 is the most common formula for finding the area, where A – lower base, b is the top base, and h is the height.

• If the height is unknown, then you can search for it using a similar formula: h = c*sin(x), where c is either AB or CD. sin(x) is the sine of the angle at any base, that is, angle DAB = angle CDA = x. Ultimately, the formula takes this form: S = (a+b)*c*sin(x)/2.
• The height can also be found using this formula:

• The final formula looks like this:

• The area of ​​an isosceles trapezoid can also be found through midline and height. The formula is: S = mh.

Let us consider the condition when a circle is inscribed in a trapezoid.

In the case shown in the picture,

QN = D = H – the diameter of the circle and at the same time the height of the trapezoid;

LO, ON, OQ = R – radii of the circle;

DC = a – upper base;

AB = b – lower base;

DAB, ABC, BCD, CDA – alpha, beta – angles of the bases of the trapezoid.

A similar case allows the area to be found using the following formulas:

• Now let's try to find the area through the diagonals and the angles between them.

In the figure we denote AC, DB – diagonals – d. Angles COB, DOB – alpha; DOC, AOB – beta. Formula for the area of ​​an isosceles trapezoid using the diagonals and the angle between them, ( S ) is:

Before finding the area of ​​a trapezoid, it is necessary to determine the known elements of the trapezoid. A trapezoid is a geometric object, namely a quadrilateral that has two parallel sides (two bases). The other two sides are lateral. If these two sides of the quadrilateral are also parallel, then it will no longer be a trapezoid, but a parallelogram. If at least one angle of a trapezoid is 90 degrees, then such a trapezoid is called rectangular. We'll look at how to find the area of ​​a rectangular trapezoid later. There is also an isosceles trapezoid, the name of which speaks for itself: the sides of such a trapezoid are equal. The distance between the bases of a trapezoid is called the height, and height is very often used to find area. The midline of a trapezoid is a segment that connects the midpoints of the sides.

Basic formulas for finding the area of ​​a trapezoid

• S= h*(a+b)/2
  Where h is the height of the trapezoid, a, b are the bases. The most commonly used formula for finding the area of ​​a trapezoid is half the sum of the bases multiplied by the height.
• S = m*h
  Where m is the midline of the trapezoid, h is the height. The area of ​​a trapezoid is also equal to the product of the midline of the trapezoid and its height.
• S=1/2*d1*d2*sin(d1^d2)
  Where d1, d2 are the diagonals of the trapezoid, sin(d1^d2) is the sine of the angle between the diagonals of the trapezoid.

There are also various formulas, derived from the basic ones, as well as a formula for calculating the area of ​​a trapezoid when all its sides are known. However, this formula is quite cumbersome and is rarely used, because, knowing all the sides of the trapezoid, you can simply determine the height or its midline. also in isosceles trapezoid you can inscribe a circle. In this case, the area of ​​the trapezoid will be calculated using the formula: 8 * radius of the circle squared.

How to find the area of ​​a rectangular trapezoid

As mentioned earlier, a trapezoid is called rectangular if it has at least one right angle. Finding the area of ​​such a trapezoid is very simple. Basically, to find the area of ​​a rectangular trapezoid, the same formulas are used as for a regular trapezoid. However, it is worth remembering that one of the sides of such a trapezoid will be the height. Also, often solving problems of finding the area of ​​a rectangular trapezoid comes down to finding the area of ​​the rectangle and triangle formed by the omitted height. Such tasks are quite simple.