Trapeze is called a quadrilateral whose only two the sides are parallel to each other.
They are called the bases of the figure, the remaining ones 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 The trapezoid is drawn at right angles from the 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: 
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.
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.
The practice of last year's Unified State Exam and State Examination shows that geometry problems cause difficulties for many schoolchildren. You can easily cope with them if you memorize all the necessary formulas and practice solving problems.
In this article you will see formulas for finding the area of a trapezoid, as well as examples of problems with solutions. You may come across the same ones in KIMs during certification exams or at Olympiads. Therefore, treat them carefully.
To begin with, let us remember that trapezoid is called a quadrilateral in which two opposite sides, also called bases, are parallel, and the other two are not.
In a trapezoid, the height (perpendicular to the base) can also be lowered. The middle line is drawn - this is a straight line that is parallel to the bases and equal to half of their sum. As well as diagonals that can intersect, forming sharp and obtuse angles. Or, in some cases, at a right angle. In addition, if the trapezoid is isosceles, a circle can be inscribed in it. And describe a circle around it.
First, let's look at the standard formulas for finding the area of a trapezoid. We will consider ways to calculate the area of isosceles and curvilinear trapezoids below.
So, imagine that you have a trapezoid with bases a and b, in which height h is lowered to the larger base. Calculating the area of a figure in this case is as easy as shelling pears. You just need to divide the sum of the lengths of the bases by two and multiply the result by the height: S = 1/2(a + b)*h.
Let's take another case: suppose in a trapezoid, in addition to the height, there is a middle line m. We know the formula for finding the length of the middle line: m = 1/2(a + b). Therefore, we can rightfully simplify the formula for the area of a trapezoid to the following type: S = m* h. In other words, to find the area of a trapezoid, you need to multiply the center line by the height.
Let's consider another option: the trapezoid contains diagonals d 1 and d 2, which do not intersect at right angles α. To calculate the area of such a trapezoid, you need to divide the product of the diagonals by two and multiply the result by the sin of the angle between them: S= 1/2d 1 d 2 *sinα.
Now consider the formula for finding the area of a trapezoid if nothing is known about it except the lengths of all its sides: a, b, c and d. It's bulky and complex formula, but it will be useful for you to remember it, just in case: S = 1/2(a + b) * √c 2 – ((1/2(b – a)) * ((b – a) 2 + c 2 – d 2)) 2.
By the way, the above examples are also true for the case when you need the formula for the area of a rectangular trapezoid. This is a trapezoid, the side of which adjoins the bases at a right angle.
A trapezoid whose sides are equal is called isosceles. We will consider several options for the formula for the area of an isosceles trapezoid.
First option: for the case when a circle with radius r is inscribed inside an isosceles trapezoid, and the side and larger base form sharp cornerα. A circle can be inscribed in a trapezoid provided that the sum of the lengths of its bases is equal to the sum of the lengths of the sides.
The area of an isosceles trapezoid is calculated as follows: multiply the square of the radius of the inscribed circle by four and divide it all by sinα: S = 4r 2 /sinα. Another area formula is a special case for the option when the angle between the large base and the side is 30 0: S = 8r2.
Second option: this time we take an isosceles trapezoid, in which in addition the diagonals d 1 and d 2 are drawn, as well as the height h. If the diagonals of a trapezoid are mutually perpendicular, the height is half the sum of the bases: h = 1/2(a + b). Knowing this, it is easy to transform the formula for the area of a trapezoid already familiar to you into this form: S = h 2.
Let's start by figuring out what a curved trapezoid is. Imagine a coordinate axis and a graph of a continuous and non-negative function f that does not change sign within a given segment on the x-axis. A curvilinear trapezoid is formed by the graph of the function y = f(x) - at the top, the x axis is at the bottom (segment), and on the sides - straight lines drawn between points a and b and the graph of the function.
It is impossible to calculate the area of such a non-standard figure using the above methods. Here you need to apply mathematical analysis and use the integral. Namely: the Newton-Leibniz formula - S = ∫ b a f(x)dx = F(x)│ b a = F(b) – F(a). In this formula, F is the antiderivative of our function on the selected segment. And the area of a curvilinear trapezoid corresponds to the increment of the antiderivative on a given segment.
To make all these formulas easier to understand in your head, here are some examples of problems for finding the area of a trapezoid. It will be best if you first try to solve the problems yourself, and only then compare the answer you receive with the ready-made solution.
Task #1: Given a trapezoid. Its larger base is 11 cm, the smaller one is 4 cm. The trapezoid has diagonals, one 12 cm long, the second 9 cm.
Solution: Construct a trapezoid AMRS. Draw a straight line РХ through vertex P so that it is parallel to the diagonal MC and intersects the straight line AC at point X. You will get a triangle APХ.
We will consider two figures obtained as a result of these manipulations: triangle APX and parallelogram CMRX.
Thanks to the parallelogram, we learn that PX = MC = 12 cm and CX = MR = 4 cm. From where we can calculate the side AX of the triangle ARX: AX = AC + CX = 11 + 4 = 15 cm.
We can also prove that the triangle APX is right-angled (to do this, apply the Pythagorean theorem - AX 2 = AP 2 + PX 2). And calculate its area: S APX = 1/2(AP * PX) = 1/2(9 * 12) = 54 cm 2.
Next you will need to prove that triangles AMP and PCX are equal in area. The basis will be the equality of the parties MR and CX (already proven above). And also the heights that you lower on these sides - they are equal to the height of the AMRS trapezoid.
All this will allow you to say that S AMPC = S APX = 54 cm 2.
Task #2: The trapezoid KRMS is given. On its lateral sides there are points O and E, while OE and KS are parallel. It is also known that the areas of trapezoids ORME and OKSE are in the ratio 1:5. RM = a and KS = b. You need to find OE.
Solution: Draw a line parallel to RK through point M, and designate the point of its intersection with OE as T. A is the point of intersection of a line drawn through point E parallel to RK with the base KS.
Let's introduce one more notation - OE = x. And also the height h 1 for the triangle TME and the height h 2 for the triangle AEC (you can independently prove the similarity of these triangles).
We will assume that b > a. The areas of the trapezoids ORME and OKSE are in the ratio 1:5, which gives us the right to create the following equation: (x + a) * h 1 = 1/5(b + x) * h 2. Let's transform and get: h 1 / h 2 = 1/5 * ((b + x)/(x + a)).
Since the triangles TME and AEC are similar, we have h 1 / h 2 = (x – a)/(b – x). Let’s combine both entries and get: (x – a)/(b – x) = 1/5 * ((b + x)/(x + a)) ↔ 5(x – a)(x + a) = (b + x)(b – x) ↔ 5(x 2 – a 2) = (b 2 – x 2) ↔ 6x 2 = b 2 + 5a 2 ↔ x = √(5a 2 + b 2)/6.
Thus, OE = x = √(5a 2 + b 2)/6.
Geometry is not the easiest of sciences, but you can certainly cope with the exam questions. It is enough to show a little perseverance in preparation. And, of course, remember all the necessary formulas.
We tried to collect all the formulas for calculating the area of a trapezoid in one place so that you can use them when you prepare for exams and revise the material.
Be sure to tell your classmates and friends about this article. in social networks. Let there be more good grades for the Unified State Examination and State Examinations!
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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.
Trapezoid is geometric figure, formed by four intersecting segments, two of which are parallel to each other and are called the bases of a 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:
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 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's another one simple formula to calculate 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 mathematics, several types of quadrilaterals are known: square, rectangle, rhombus, parallelogram. Among them is a trapezoid - a type of convex quadrilateral in which two sides are parallel and the other two are not. The parallel opposite sides are called the bases, and the other two are called the lateral sides of the trapezoid. The segment that connects the midpoints of the sides is called the midline. There are several types of trapezoids: isosceles, rectangular, curved. For each type of trapezoid there are formulas for finding the area.
To find the area of a trapezoid, you need to know the length of its bases and height. The height of a trapezoid is a segment perpendicular to the bases. Let the top base be a, the bottom base be b, and the height be h. Then you can calculate the area S using the formula:
S = ½ * (a+b) * h
those. take half the sum of the bases multiplied by the height.
It will also be possible to calculate the area of the trapezoid if the height and center line are known. Let's denote the middle line - m. Then
Let's solve a more complicated problem: the lengths of the four sides of the trapezoid are known - a, b, c, d. Then the area will be found using the formula:
If the lengths of the diagonals and the angle between them are known, then the area is searched as follows:
S = ½ * d1 * d2 * sin α
where d with indices 1 and 2 are diagonals. In this formula, the sine of the angle is given in the calculation.
Given the known lengths of the bases a and b and two angles at the lower base, the area is calculated as follows:
S = ½ * (b2 - a2) * (sin α * sin β / sin(α + β))
An isosceles trapezoid is a special case of a trapezoid. Its difference is that such a trapezoid is convex quadrilateral with an axis of symmetry passing through the midpoints of two opposite sides. Its sides are equal.
There are several ways to find the area of an isosceles trapezoid.
S = c * sin α * (a + c * cos α)
where a is the top base, c is the side.
S = c * sin α * (b – c * cos α)
S = ½ * (b2 – a2) * tan α
S = ½ * d2 * sin α
Let the lateral side be c, the middle line be m, and the angle be a, then:
S = m * c * sin α
Sometimes you can inscribe a circle in an equilateral trapezoid, the radius of which will be r.
It is known that a circle can be inscribed in any trapezoid if the sum of the lengths of the bases is equal to the sum of the lengths of its sides. Then the area can be found through the radius of the inscribed circle and the angle at the lower base:
S = 4r2 / sin α
The same calculation is made using the diameter D of the inscribed circle (by the way, it coincides with the height of the trapezoid):
Knowing the base and angle, the area of an isosceles trapezoid is calculated as follows:
S = a * b / sin α
(this and subsequent formulas are valid only for trapezoids with an inscribed circle).
Using the bases and radius of the circle, the area is found as follows:
If only the bases are known, then the area is calculated using the formula:
Through the bases and the side line, the area of the trapezoid with the inscribed circle and through the bases and the middle line - m is calculated as follows:
Area of a rectangular trapezoid
A trapezoid is called rectangular if one of its sides is perpendicular to the base. In this case, the length of the side coincides with the height of the trapezoid.
A rectangular trapezoid consists of a square and a triangle. Having found the area of each of the figures, add up the results and get total area figures.
Also suitable for calculating the area of a rectangular trapezoid general formulas to calculate the area of a trapezoid.
S = (a + b) * h / 2
The side side c can act as h (height). Then the formula looks like this:
S = (a + b) * c / 2
or by the length of the lateral perpendicular side:
S = ½ * d1 * d2 * sin α
If the diagonals are perpendicular, then the formula simplifies to:
S = ½ * d1 * d2
This formula is valid for bases. If we take the lengths of the sides, then one of them will be equal to twice the radius. The formula will look like this:
S = (2r + c) * r
where m is the length of the center line.
A curvilinear trapezoid is a flat figure bounded by the graph of a non-negative continuous function y = f(x), defined on the segment, the abscissa axis and the straight lines x = a, x = b. Essentially, two of its sides are parallel to each other (the bases), the third side is perpendicular to the bases, and the fourth is a curve corresponding to the graph of the function.
The area of a curvilinear trapezoid is sought through the integral using the Newton-Leibniz formula:
This is how areas are calculated various types trapezoid. But, in addition to the properties of the sides, trapezoids have identical properties corners Like all existing quadrilaterals, the sum internal corners a trapezoid equals 360 degrees. And the sum of the angles adjacent to the side is 180 degrees.
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 classic look The trapezoid is depicted as follows: the larger base is at the bottom, and the smaller base 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.
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 get a rectangle in area equal to area 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.
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That's all. Good luck to you!
Sincerely, Alexander.
