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: 
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 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.
Trapezoid is special kind a 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 - distance between parallel bases figures;
- middle line - a segment connecting the midpoints of the sides.
This geometric figure 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 decided 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 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.
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 indicated by a pair of capital letters Latin letters corresponding to the vertices that the segments connect.
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 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. Square 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.
Behind 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 known meaning angle at the base according to 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 (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.
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.
What you need to know about the trapezoid?
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 acute and obtuse angles. Or in in some cases, at right angles. In addition, if the trapezoid is isosceles, a circle can be inscribed in it. And describe a circle around it.
Trapezoid area formulas
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 form: 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. This is a cumbersome 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.
Isosceles trapezoid
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 an acute angle α. 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'll take 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.
Formula for the area of a curved trapezoid
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.
Sample problems
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.
Conclusion
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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