Instructions
To make both methods more understandable, we can give a couple of examples.
Example 1: the length of the midline of the trapezoid is 10 cm, its area is 100 cm². To find the height of this trapezoid, you need to do:
h = 100/10 = 10 cm
Answer: the height of this trapezoid is 10 cm
Example 2: the area of the trapezoid is 100 cm², the lengths of the bases are 8 cm and 12 cm. To find the height of this trapezoid, you need to perform the following action:
h = (2*100)/(8+12) = 200/20 = 10 cm
Answer: the height of this trapezoid is 20 cm
note
There are several types of trapezoids:
An isosceles trapezoid is a trapezoid in which the sides are equal to each other.
A right-angled trapezoid is a trapezoid with one of its interior angles measuring 90 degrees.
It is worth noting that in a rectangular trapezoid the height coincides with the length of the side when right angle.
You can describe a circle around a trapezoid, or fit it inside a given figure. You can inscribe a circle only if the sum of its bases is equal to the sum of its opposite sides. A circle can only be described around isosceles trapezoid.
A parallelogram is a special case of a trapezoid, because the definition of a trapezoid does not in any way contradict the definition of a parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel to each other. For a trapezoid, the definition refers only to a pair of its sides. Therefore, any parallelogram is also a trapezoid. The reverse statement is not true.
Sources:
- how to find the area of a trapezoid formula
Tip 2: How to find the height of a trapezoid if the area is known
A trapezoid is a quadrilateral in which two of its four sides are parallel to each other. The parallel sides are the bases of the given one, the other two are the lateral sides of the given one. trapezoids. Find height trapezoids, if known square, it will be very easy.
Instructions
You need to figure out how to calculate square original trapezoids. There are several formulas for this, depending on the initial data: S = ((a+b)*h)/2, where a and b are bases trapezoids, and h is its height (Height trapezoids- perpendicular, lowered from one base trapezoids to another);
S = m*h, where m is line trapezoids(The middle line is a segment with bases trapezoids and connecting the midpoints of its sides).
To make it clearer, similar problems can be considered: Example 1: Given a trapezoid with square 68 cm², the middle line of which is 8 cm, you need to find height given trapezoids. In order to solve this problem, you need to use the previously derived formula:
h = 68/8 = 8.5 cm Answer: height of this trapezoids is 8.5 cmExample 2: Let y trapezoids square equals 120 cm², the length of the bases of this trapezoids 8 cm and 12 cm respectively, you need to find height this trapezoids. To do this, you need to apply one of the derived formulas:
h = (2*120)/(8+12) = 240/20 = 12 cmAnswer: given height trapezoids equal to 12 cm
Video on the topic
note
Any trapezoid has a number of properties:
The midline of a trapezoid is equal to half the sum of its bases;
The segment that connects the diagonals of a trapezoid is equal to half the difference of its bases;
If a straight line is drawn through the midpoints of the bases, then it will intersect the point of intersection of the diagonals of the trapezoid;
A circle can be inscribed in a trapezoid if the sum of the bases of the trapezoid is equal to the sum of its sides.
Use these properties when solving problems.
Tip 3: How to find the area of a trapezoid if the bases are known
By geometric definition, a trapezoid is a quadrilateral with only one pair of sides parallel. These sides are hers reasons. Distance between reasons called height trapezoids. Find square trapezoids possible using geometric formulas.
Instructions
Measure the bases and trapezoids ABCD. Usually they are given in tasks. Let in this example problem the base AD (a) trapezoids will be equal to 10 cm, base BC (b) - 6 cm, height trapezoids BK (h) - 8 cm. Use geometric to find area trapezoids, if the lengths of its bases and heights are known - S= 1/2 (a+b)*h, where: - a - the size of the base AD trapezoids ABCD, - b - the value of the base BC, - h - the value of the height BK.
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 (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:
- The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.
- Isosceles trapezoids have equal sides and the angles they form with the bases.
- The midpoints of the diagonals of a trapezoid and the point of intersection of its diagonals are on the same straight line.
- 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
- 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.
- An isosceles trapezoid can be described by a circle. And vice versa. If a trapezoid fits into a circle, then it is isosceles.
- 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 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.
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 will be analyzed various ways Finding 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. 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 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, 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. Dana 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 Peak 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 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 tests on this topic.
Area of a trapezoid. Greetings! In this publication we will look at this formula. Why is she exactly like this and how to understand her. If there is understanding, then you don’t need to teach it. If you just want to look at this formula and urgently, then you can immediately scroll down the page))
Now in detail and in order.
A trapezoid is a quadrilateral, two sides of this quadrilateral are parallel, the other two are not. Those that are not parallel are the bases of the trapezoid. The other two are called sides.
If the sides are equal, then the trapezoid is called isosceles. If one of the sides is perpendicular to the bases, then such a trapezoid is called rectangular.
In its classic form, a trapezoid is depicted as follows - the larger base is at the bottom, respectively, the smaller one is at the top. But no one forbids depicting her and vice versa. Here are the sketches:
Next important concept.
The midline of a trapezoid is a segment that connects the midpoints of the sides. The middle line is parallel to the bases of the trapezoid and equal to their half-sum.
Now let's delve deeper. Why is this so?
Consider a trapezoid with bases a and b and with the middle line l, and perform some additional constructions: draw straight lines through the bases, and perpendiculars through the ends of the midline until they intersect with the bases:
*Letter designations for vertices and other points are not included intentionally to avoid unnecessary designations.
Look, triangles 1 and 2 are equal according to the second sign of equality of triangles, triangles 3 and 4 are the same. From the equality of triangles follows the equality of the elements, namely the legs (they are indicated in blue and red, respectively).
Now attention! If we mentally “cut off” the blue and red segments from the lower base, then we will be left with a segment (this is the side of the rectangle) equal to the middle line. Next, if we “glue” the cut blue and red segments to the upper base of the trapezoid, then we will also get a segment (this is also the side of the rectangle) equal to the midline of the trapezoid.
Got it? It turns out that the sum of the bases will be equal to the two middle lines of the trapezoid:
View another explanation
Let's do the following - construct a straight line passing through the lower base of the trapezoid and a straight line that will pass through points A and B:
We get triangles 1 and 2, they are equal along the side and adjacent angles (the second sign of equality of triangles). This means that the resulting segment (in the sketch it is indicated in blue) is equal to the upper base of the trapezoid.
Now consider the triangle:
*The midline of this trapezoid and the midline of the triangle coincide.
It is known that a triangle is equal to half of the base parallel to it, that is:
Okay, we figured it out. Now about the area of the trapezoid.
Trapezoid area formula:
They say: the area of a trapezoid is equal to the product of half the sum of its bases and height.
That is, it turns out that it is equal to the product of the center line and the height:
You've probably already noticed that this is obvious. Geometrically, this can be expressed this way: if we mentally cut off triangles 2 and 4 from the trapezoid and place them on triangles 1 and 3, respectively:
Then we will get a rectangle with an area equal to the area of our trapezoid. The area of this rectangle will be equal to the product of the center line and the height, that is, we can write:
But the point here is not in writing, of course, but in understanding.
Download (view) article material in *pdf format
That's all. Good luck to you!
Sincerely, Alexander.
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 general formula 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 performing 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 is it 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 corner at vertex B equal to angle between the diagonals of a 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 came across 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 the trapezoid is doubled more area 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). 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 didn't choose the best complex tasks(don’t be afraid of mechanics and mathematics!) with the expectation of the possibility of them independent decision. Decide for your health! If you need preparation for the Unified State Exam in mathematics, then without participation in this process, formulas for the area of a trapezoid may arise serious problems 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.
