Calculate the area of ​​a trapezoid online on four sides. Area of ​​trapezoid

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. This article will analyze in accessible language various ways to find the area of ​​a trapezoid. 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. There are different types of trapezoid: 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 problems are the most common in the school geometry course 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

A special case of a rectangular trapezoid is known. 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, it has a 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 in which two sides are the same. This means that the other side of the triangle is equal to the height, that is, 16 cm. All that remains is 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 Peak formula

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 fewer errors and better the 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 assignments on this topic.

This calculator has calculated 2192 problems on the topic "Area of ​​a trapezoid"

AREA OF TRAPEZOID

Choose the formula for calculating the area of ​​a trapezoid that you plan to use to solve the problem assigned to you:

General theory for calculating the area of ​​a trapezoid.

Trapezoid - This is a flat figure consisting of four points, three of which do not lie on the same line, and four segments (sides) connecting these four points in pairs, in which two opposite sides are parallel (lie on parallel lines), and the other two are not parallel.

The points are called vertices of a trapezoid and are indicated in capital Latin letters.

The segments are called trapezoid sides and are designated by a pair of capital Latin letters corresponding to the vertices that connect the segments.

Two parallel sides of a trapezoid are called trapezoid bases .

Two non-parallel sides of a trapezoid are called sides of the trapezoid .

Figure No. 1: Trapezoid ABCD

Figure No. 1 shows the trapezoid ABCD with vertices A, B, C, D and sides AB, BC, CD, DA.

AB ǁ DC - bases of trapezoid ABCD.

AD, BC - lateral sides of the trapezoid ABCD.

The angle formed by rays AB and AD is called the angle at vertex A. It is denoted as ÐA or ÐBAD, or ÐDAB.

The angle formed by rays BA and BC is called the angle at vertex B. It is denoted as ÐB or ÐABC, or ÐCBA.

The angle formed by rays CB and CD is called the vertex angle C. It is denoted as ÐC or ÐDCB, or ÐBCD.

The angle formed by rays AD and CD is called the vertex angle D. It is denoted as ÐD or ÐADC, or ÐCDA.

Figure No. 2: Trapezoid ABCD

In Figure 2, the segment MN connecting the midpoints of the lateral sides is called midline of the trapezoid.

Midline of trapezoid parallel to the bases and equal to their half-sum. That is, .

Figure No. 3: Isosceles trapezoid ABCD

In Figure 3, AD=BC.

The trapezoid is called isosceles (isosceles), if its sides are equal.

Figure No. 4: Rectangular trapezoid ABCD

In Figure No. 4, angle D is straight (equal to 90°).

The trapezoid is called rectangular, if the angle at the side is straight.

Area S flat figures, which include the trapezoid, are called limited closed space on a plane. The area of ​​a flat figure shows the size of this figure.

The area has several properties:

1. It cannot be negative.

2. If a certain closed area on the plane is given, which is made up of several figures that do not intersect each other (that is, the figures do not have common internal points, but may well touch each other), then the area of ​​such an area is equal to the sum of the areas of its constituent figures .

3. If two figures are equal, then their areas are equal.

4. The area of ​​a square, which is built on a unit segment, is equal to one.

For unit measurements area take the area of ​​a square whose side is equal to unit measurements segments.

When solving problems, the following formulas for calculating the area of ​​a trapezoid are often used:

1. The area of ​​a trapezoid is equal to half the sum of its bases multiplied by its height:

2. The area of ​​a trapezoid is equal to the product of its midline and its height:

3. With known lengths of the bases and sides of the trapezoid, its area can be calculated using the formula:

4. It is possible to calculate the area of ​​an isosceles trapezoid with a known length of the radius of the circle inscribed in the trapezoid and a known value of the angle at the base using the following formula:

Example 1: Calculate the area of ​​a trapezoid with bases a=7, b=3 and height h=15.

Solution:

Answer:

Example 2: Find the side of the base of a trapezoid with area S = 35 cm 2, height h = 7 cm and second base b = 2 cm.

Solution:

To find the side of the base of a trapezoid, we use the formula for calculating the area:

Let us express from this formula the side of the base of the trapezoid:

Thus, we have the following:

Answer:

Example 3: Find the height of a trapezoid with area S = 17 cm 2 and bases a = 30 cm, b = 4 cm.

Solution:

To find the height of a trapezoid, we use the formula for calculating the area:

Thus, we have the following:

Answer:

Example 4: Calculate the area of ​​a trapezoid with height h=24 and center line m=5.

Solution:

To find the area of ​​a trapezoid, we use the following formula for calculating the area:

Thus, we have the following:

Answer:

Example 5: Find the height of a trapezoid with area S = 48 cm 2 and center line m = 6 cm.

Solution:

To find the height of a trapezoid, we use the formula for calculating the area of ​​a trapezoid:

Let us express the height of the trapezoid from this formula:

Thus, we have the following:

Answer:

Example 6: Find the midline of a trapezoid with area S = 56 and height h=4.

Solution:

To find the midline of a trapezoid, we use the formula for calculating the area of ​​a trapezoid:

Let us express the middle line of the trapezoid from this formula:

Thus, we have the following.

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 the construction of modern apartments, or in design renovation 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 (equilateral) 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 also decompose any trapezoid into simpler 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 is more suitable for mathematics problems than for everyday problems, 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.

A trapezoid is a special type of quadrilateral in which two opposite sides are parallel to each other, but the other two are not. Various real objects have a trapezoidal shape, so you may need to calculate the perimeter of such a geometric figure to solve everyday or school problems.

Trapezoid geometry

A trapezoid (from the Greek “trapezion” - table) is a figure on a plane limited by four segments, two of which are parallel and two are not. Parallel segments are called the bases of the trapezoid, and non-parallel segments are called the sides of the figure. The sides and their angles of inclination determine the type of trapezoid, which can be scalene, isosceles or rectangular. In addition to the bases and sides, the trapezoid has two more elements:

• height - the distance between the parallel bases of the figure;
• middle line - a segment connecting the midpoints of the sides.

This geometric figure is widespread in real life.

Trapezoid in reality

In everyday life, many real objects take a trapezoidal shape. You can easily find trapezoids in the following areas of human activity:

• interior design and decor - sofas, tabletops, walls, carpets, suspended ceilings;
• landscape design - boundaries of lawns and artificial reservoirs, forms of decorative elements;
• fashion - the form of clothing, shoes and accessories;
• architecture - windows, walls, building foundations;
• production - various products and parts.

With such widespread use of trapezoids, specialists often have to calculate the perimeter of a geometric figure.

Trapezoid perimeter

The perimeter of a figure is a numerical characteristic that is calculated as the sum of the lengths of all sides of the n-gon. A trapezoid is a quadrilateral and in general all its sides have different lengths, so the perimeter is calculated using the formula:

P = a + b + c + d,

where a and c are the bases of the figure, b and d are its sides.

Although we don't need to know the height when calculating the perimeter of a trapezoid, the calculator code requires entering this variable. Since height has no effect on calculations, when using our online calculator you can enter any height value that is greater than zero. Let's look at a couple of examples.

Real life examples

Handkerchief

Let's say you have a trapezoid-shaped scarf and you want to trim it with fringe. You will need to know the perimeter of the scarf so you don't buy extra material or go to the store twice. Let your isosceles scarf have the following parameters: a = 120 cm, b = 60 cm, c = 100 cm, d = 60 cm. We enter these data into the online form and get the answer in the form:

Thus, the perimeter of the scarf is 340 cm, and this is exactly the length of the fringe braid to finish it.

Slopes

For example, you decide to make slopes for non-standard metal-plastic windows that have a trapezoidal shape. Such windows are widely used in building design, creating a composition of several sashes. Most often, such windows are made in the form of a rectangular trapezoid. Let's find out how much material is needed to make the slopes of such a window. A standard window has the following parameters a = 140 cm, b = 20 cm, c = 180 cm, d = 50 cm. We use these data and get the result in the form

Therefore, the perimeter of the trapezoidal window is 390 cm, and that is exactly how many plastic panels you will need to buy to form the slopes.

Conclusion

The trapezoid is a popular figure in everyday life, the determination of whose parameters may be needed in the most unexpected situations. Calculating trapezoidal perimeters is necessary for many professionals: from engineers and architects to designers and mechanics. Our catalog of online calculators will allow you to perform calculations for any geometric shapes and bodies.

There are many ways to find the area of ​​a trapezoid. Usually a math tutor knows several methods of calculating it, let’s look at them in more detail:
1) , where AD and BC are the bases, and BH is the height of the trapezoid. Proof: draw the diagonal BD and express the areas of triangles ABD and CDB through the half product of their bases and heights:

, where DP is the external height in

Let us add these equalities term by term and taking into account that the heights BH and DP are equal, we obtain:

Let's put it out of brackets

Q.E.D.

Corollary to the formula for the area of ​​a trapezoid:
Since the half-sum of the bases is equal to MN - the midline of the trapezoid, then

2) Application of the general formula for the area of ​​a quadrilateral.
The area of ​​a quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them
To prove it, it is enough to divide the trapezoid into 4 triangles, express the area of ​​each in terms of “half the product of the diagonals and the sine of the angle between them” (taken as the angle, add the resulting expressions, take them out of the bracket and factor this bracket using the grouping method to obtain its equality to the expression. Hence

3) Diagonal shift method
This is my name. A math tutor will not come across such a heading in school textbooks. A description of the technique can only be found in additional textbooks as an example of solving a problem. I would like to note that most of the interesting and useful facts about planimetry are revealed to students by math tutors in the process of doing practical work. This is extremely suboptimal, because the student needs to isolate them into separate theorems and call them “big names.” One of these is “diagonal shift”. What are we talking about? Let us draw a line parallel to AC through vertex B until it intersects with the lower base at point E. In this case, the quadrilateral EBCA will be a parallelogram (by definition) and therefore BC=EA and EB=AC. The first equality is important to us now. We have:

Note that the triangle BED, whose area is equal to the area of ​​the trapezoid, has several more remarkable properties:
1) Its area is equal to the area of ​​the trapezoid
2) Its isosceles occurs simultaneously with the isosceles of the trapezoid itself
3) Its upper angle at vertex B is equal to the angle between the diagonals of the trapezoid (which is very often used in problems)
4) Its median BK is equal to the distance QS between the midpoints of the bases of the trapezoid. I recently encountered the use of this property when preparing a student for Mechanics and Mathematics at Moscow State University using Tkachuk’s textbook, 1973 version (the problem is given at the bottom of the page).

Special techniques for a math tutor.

Sometimes I propose problems using a very tricky way of finding the area of ​​a trapezoid. I classify it as a special technique because in practice the tutor uses them extremely rarely. If you need preparation for the Unified State Exam in mathematics only in Part B, you don’t have to read about them. For others, I'll tell you further. It turns out that the area of ​​a trapezoid is twice the area of ​​a triangle with vertices at the ends of one side and the middle of the other, that is, the ABS triangle in the figure:
Proof: draw the heights SM and SN in triangles BCS and ADS and express the sum of the areas of these triangles:

Since point S is the midpoint of CD, then (prove it yourself). Let’s find the sum of the areas of the triangles:

Since this sum turned out to be equal to half the area of ​​the trapezoid, then its second half. Etc.

I would include in the tutor’s collection of special techniques the form of calculating the area of ​​an isosceles trapezoid along its sides: where p is the semi-perimeter of the trapezoid. I won't give proof. Otherwise, your math tutor will be left without a job :). Come to class!

Problems on the area of ​​a trapezoid:

Math tutor's note: The list below is not a methodological accompaniment to the topic, it is only a small selection of interesting tasks based on the techniques discussed above.

1) The lower base of an isosceles trapezoid is 13, and the upper is 5. Find the area of ​​the trapezoid if its diagonal is perpendicular to the side.
2) Find the area of ​​a trapezoid if its bases are 2cm and 5cm, and its sides are 2cm and 3cm.
3) In an isosceles trapezoid, the larger base is 11, the side is 5, and the diagonal is Find the area of ​​the trapezoid.
4) The diagonal of an isosceles trapezoid is 5 and the midline is 4. Find the area.
5) In an isosceles trapezoid, the bases are 12 and 20, and the diagonals are mutually perpendicular. Calculate the area of ​​a trapezoid
6) The diagonal of an isosceles trapezoid makes an angle with its lower base. Find the area of ​​the trapezoid if its height is 6 cm.
7) The area of ​​the trapezoid is 20, and one of its sides is 4 cm. Find the distance to it from the middle of the opposite side.
8) The diagonal of an isosceles trapezoid divides it into triangles with areas of 6 and 14. Find the height if the lateral side is 4.
9) In a trapezoid, the diagonals are equal to 3 and 5, and the segment connecting the midpoints of the bases is equal to 2. Find the area of ​​the trapezoid (Mekhmat MSU, 1970).

I chose not the most difficult problems (don’t be afraid of mechanical engineering!) with the expectation that I would be able to solve them independently. Decide for your health! If you need preparation for the Unified State Exam in mathematics, then without the participation of the formula for the area of ​​a trapezoid in this process, serious problems may arise even with problem B6 and even more so with C4. Do not start the topic and in case of any difficulties, ask for help. A math tutor is always happy to help you.

Kolpakov A.N.
Mathematics tutor in Moscow, preparation for the Unified State Exam in Strogino.