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 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 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 through “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 triangle ABS 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 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.
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, curvilinear. For each type of trapezoid there are formulas for finding the area.
Area of trapezoid
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 found 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))
Area of an isosceles trapezoid
An isosceles trapezoid is a special case of a trapezoid. Its difference is that such a trapezoid is a 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.
- Through the lengths of three sides. In this case, the lengths of the sides will coincide, therefore they are designated by one value - c, and a and b - the lengths of the bases:
- If the length of the upper base, the side and the angle at the lower base are known, then the area is calculated as follows:
S = c * sin α * (a + c * cos α)
where a is the top base, c is the side.
- If instead of the upper base the length of the lower one is known - b, the area is calculated using the formula:
S = c * sin α * (b – c * cos α)
- If, when two bases and the angle at the lower base are known, the area is calculated through the tangent of the angle:
S = ½ * (b2 – a2) * tan α
- The area is also calculated through the diagonals and the angle between them. In this case, the diagonals are equal in length, so we denote each by the letter d without subscripts:
S = ½ * d2 * sin α
- Let's calculate the area of the trapezoid, knowing the length of the side, the center line and the angle at the bottom base.
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 the total area of the figure.
Also, general formulas for calculating the area of a trapezoid are suitable for calculating the area of a rectangular trapezoid.
- If the lengths of the bases and the height (or the perpendicular side side) are known, then the area is calculated using the formula:
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
- Another way to calculate area is to multiply the length of the center line by the height:
or by the length of the lateral perpendicular side:
- The next way to calculate is through half the product of the diagonals and the sine of the angle between them:
S = ½ * d1 * d2 * sin α
If the diagonals are perpendicular, then the formula simplifies to:
S = ½ * d1 * d2
- Another way to calculate is through the semi-perimeter (the sum of the lengths of two opposite sides) and the radius of the inscribed circle.
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
- If a circle is inscribed in a trapezoid, then the area is calculated in the same way:
where m is the length of the center line.
Area of a curved trapezoid
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 the areas of various types of trapezoids are calculated. But, in addition to the properties of the sides, trapezoids have the same properties of angles. Like all existing quadrilaterals, the sum of the interior angles of a trapezoid is 360 degrees. And the sum of the angles adjacent to the side is 180 degrees.
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:
- 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 Examination 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 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.
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 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 that is 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.
Examples of 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 size. 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 the RK through point M, and designate the point of its intersection with OE as T. A is the point of intersection of the line drawn through point E parallel to the 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 on social networks about this article. Let there be more good grades for the Unified State Examination and State Examinations!
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