The practice of last year's USE and GIA shows that geometry problems cause difficulties for many students. You can easily cope with them if you memorize everything necessary formulas and practice problem solving.
In this article, you will see formulas for finding the area of a trapezoid, as well as examples of problems with solutions. The same ones can come across to you in KIMs at certification exams or at olympiads. Therefore, treat them carefully.
What you need to know about the trapezoid?
To begin with, let's remember that trapeze a quadrilateral is called, in which two opposite sides, they are also called bases, are parallel, and the other two are not.
In a trapezoid, the height (perpendicular to the base) can also be omitted. 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 individual 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.
Trapezium area formulas
To begin, consider standard formulas finding the area of a trapezoid. Ways to calculate the area of isosceles and curvilinear trapezoids will be considered below.
So, imagine that you have a trapezoid with bases a and b, in which the height h is lowered to the larger base. Calculating the area of a figure in this case is easy. You just need to divide by two the sum of the lengths of the bases and multiply what happens by the height: S = 1/2(a + b)*h.
Let's take another case: suppose that in addition to the height, the trapezoid has a median line m. We know the formula for finding the length middle line: m = 1/2(a + b). Therefore, we can rightfully simplify the formula for the area of a trapezoid to the following kind: S = m * h. In other words, to find the area of a trapezoid, you need to multiply the midline by the height.
Let's consider one more option: diagonals d 1 and d 2 are drawn in a trapezoid, which intersect not at a right angle α. To calculate the area of such a trapezoid, you need to halve the product of the diagonals and multiply what you get 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 complicated formula, but it will be useful for you to remember it just in case: S \u003d 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 area formula 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 variants of the area formula isosceles trapezium.
First option: for the case when a circle with radius r is inscribed inside an isosceles trapezoid, and the lateral side and the larger base form sharp corner a. 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.
The 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 convert the trapezoid area formula already familiar to you into this form: S = h2.
The formula for the area of a curvilinear trapezoid
Let's start by understanding: what is a curvilinear trapezoid. 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 \u003d f (x) - at the top, the x axis - 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 interval. And the square curvilinear trapezoid corresponds to the increment of the antiderivative on the given interval.
Task examples
To make all these formulas better in your head, here are some examples of tasks for finding the area of a trapezoid. It would be best if you first try to solve the problems yourself, and only then check the answer you received with the ready-made solution.
Task #1: Given a trapezoid. Its larger base is 11 cm, the smaller one is 4 cm. The trapezium has diagonals, one 12 cm long, the other 9 cm long.
Solution: Build a trapezoid AMRS. Draw line RX through vertex P so that it is parallel to diagonal MC and intersects line AC at point X. You get triangle APX.
We will consider two figures obtained as a result of these manipulations: the triangle APX and the parallelogram CMPX.
Thanks to the parallelogram, we learn that PX = MC = 12 cm and CX = MP = 4 cm. Where can we calculate the side AX of the triangle ARCH: AX \u003d AC + CX \u003d 11 + 4 \u003d 15 cm.
We can also prove that the triangle ARCH is right-angled (to do this, apply the Pythagorean theorem - AX 2 \u003d AP 2 + PX 2). And calculate its area: S APX \u003d 1/2 (AP * PX) \u003d 1/2 (9 * 12) \u003d 54 cm 2.
Next, you need to prove that the triangles AMP and PCX are equal in area. The basis will be the equality of the sides MP 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 assert that S AMPC \u003d S APX \u003d 54 cm 2.
Task #2: Given a trapezoid KRMS. Points O and E are located on its lateral sides, while OE and KS are parallel. It is also known that the areas of the trapezoid ORME and OXE are in the ratio 1:5. PM = a and KS = b. You need to find an OE.
Solution: Draw a line through point M parallel to RK, and designate the point of its intersection with OE as T. A - the point of intersection of the line drawn through point E parallel to RK with the base of KS.
Let's introduce one more notation - OE = x. As well as the height h 1 for the TME triangle and the height h 2 for the AEC triangle (you can prove the similarity of these triangles yourself).
We will assume that b > a. The areas of the trapezoids ORME and OXE are related as 1:5, which gives us the right to draw up the following equation: (x + a) * h 1 \u003d 1/5 (b + x) * h 2. Let's transform and get: h 1 / h 2 \u003d 1/5 * ((b + x) / (x + a)).
Since the triangles TME and AEC are similar, we have h 1 / h 2 = (x - a) / (b - x). Combine both entries and get: (x - a) / (b - x) \u003d 1/5 * ((b + x) / (x + a)) ↔ 5 (x - a) (x + a) \u003d (b + x) (b - x) ↔ 5 (x 2 - a 2) \u003d (b 2 - x 2) ↔ 6x 2 \u003d b 2 + 5a 2 ↔ x \u003d √ (5a 2 + b 2) / 6.
Thus, OE \u003d x \u003d √ (5a 2 + b 2) / 6.
Conclusion
Geometry is not the easiest of the sciences, but you will certainly be able to cope with exam tasks. It just takes a little patience in preparation. And, of course, remember all the necessary formulas.
We tried to collect in one place all the formulas for calculating the area of a trapezoid so that you can use them when you prepare for exams and repeat the material.
Be sure to tell your classmates and friends about this article in in social networks. Let good grades there will be more for the USE and GIA!
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General theory for calculating the area of a trapezoid.
Trapeze - this is a flat figure consisting of four points, three of which do not lie on one straight 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 tops of a trapezoid and are denoted by capital Latin letters.
The segments are called sides of a trapezoid and are denoted by a pair of capital Latin letters corresponding to the vertices that the segments connect.
The two parallel sides of a trapezoid are called bases of a trapezoid .
Two non-parallel sides of a trapezoid are called sides of a trapezoid .
Figure #1: Trapezium ABCD
Figure 1 shows the trapezoid ABCD with vertices A,B,C, D and sides AB, BC, CD, DA.
AB ǁ DC - bases of the trapezoid ABCD.
AD, BC are the sides of the trapezoid ABCD.
The angle formed by the rays AB and AD is called the angle at the vertex A. It is denoted as ÐA or ÐBAD, or ÐDAB.
The angle formed by the rays BA and BC is called the angle at the vertex B. It is designated as ÐB or ÐABC, or ÐCBA.
The angle formed by rays CB and CD is called apex angle C. It is denoted as ÐC or ÐDCB or ÐBCD.
The angle formed by the rays AD and CD is called the vertex angle D. It is denoted as ÐD or ÐADC or ÐCDA.
Figure #2: Trapezium ABCD
In Figure 2, the segment MN connecting the midpoints of the sides is called midline of the trapezium.
Median line of the trapezoid parallel to the bases and equal to their half-sum. That is, 
.
Figure #3: Isosceles trapezoid ABCD
In Figure #3, AD=BC.
The trapezoid is called isosceles (isosceles) if its sides are equal.
Figure #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 lateral side is straight.
Square S flat figures, to which the trapezoid also belongs, is called a bounded closed space on a plane. The area of a flat figure shows the size of this figure.
The area has several properties:
1. It cannot be negative.
2. If some closed area on a plane is given, which is composed of several figures that do not intersect with each other (that is, the figures do not have common internal points, but may well touch each other), then the area of \u200b\u200bsuch 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 built on a unit segment is equal to one.
Per 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 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 height:
3. With known lengths of the bases and sides of the trapezoid, its area can be calculated by 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 value 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 the trapezoid, we use the formula for calculating the area:
We 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 cm2 and bases a=30 cm, b=4 cm.
Solution:
To find the height of the 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 midline m=5.
Solution:
To find the area of a trapezoid, 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 midline m = 6 cm.
Solution:
To find the height of a trapezoid, we use the formula for calculating the area of a trapezoid:
We 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:
We express from this formula the midline of the trapezoid:
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 in everyday life it occurs very rarely, but sometimes it turns out to be necessary, for example, to find the area of \u200b\u200ba room in the form of a trapezoid, which are increasingly used in construction modern apartments, or in design projects for repairs.
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 on. 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, a trapezoid has particular types in the form of an isosceles (isosceles) 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.
Trapezoids have some interesting properties:
- The midline of a trapezoid is half the sum of the bases and parallel to them.
- Isosceles trapeziums have equal sides and angles that 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 is inscribed in 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 a trapezoid will be half the sum of its bases multiplied by its height. In the form of a formula, 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, a trapezoid using the midline 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 more simple figures: a rectangle and one or two triangles, and if it’s easier for you, then find the area of \u200b\u200bthe 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 the trapezoid is equal to the product of its midline and the height of the trapezoid and is written as: S \u003d 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 math problems than for everyday problems, since in real conditions you will not know the length of the midline 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 \u003d ((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 more ways to find the area of a trapezoid. But, they are about as inconvenient as the last formula, which means it makes no sense to dwell on them. Therefore, we recommend that you use the first formula from the article and wish you always get accurate results.
In mathematics, several types of quadrilaterals are known: square, rectangle, rhombus, parallelogram. Among them is a trapezoid - a kind 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 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.
Trapezium area
To find the area of a trapezoid, you need to know the length of its bases and its 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 by the formula:
S = ½ * (a + b) * h
those. take half the sum of the bases multiplied by the height.
You can also calculate the area of a trapezoid if you know the value of the height and the midline. Let's denote the middle line - m. Then
Let's solve the problem more complicated: we know the lengths of the four sides of the trapezoid - a, b, c, d. Then the area is found by the formula:
If the lengths of the diagonals and the angle between them are known, then the area is sought as follows:
S = ½ * d1 * d2 * sinα
where d with indices 1 and 2 are diagonals. In this formula, the sine of the angle is given in the calculation.
With known base lengths 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 special case trapezoid. Its difference is that such a trapezoid is a convex quadrangle 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 match, therefore they are indicated by one value - c, a and b - the lengths of the bases:
- If the length of the upper base, lateral 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 upper base, c is the side.
- If instead of the upper base, the length of the lower base is known - b, the area is calculated by 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 using the tangent of the angle:
S = ½ * (b2 - a2) * tg α
- Also, the area is calculated through the diagonals and the angle between them. In this case, the diagonals are equal in length, so each is denoted by the letter d without indices:
S = ½ * d2 * sinα
- Calculate the area of the trapezoid, knowing the length of the lateral side, the midline and the angle at the lower base.
Let the side - c, the middle line - m, the corner - a, then:
S = m * c * sinα
Sometimes a circle can be inscribed 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 is found through the radius of the inscribed circle and the angle at the lower base:
S = 4r2 / sinα
The same calculation is made through the diameter D of the inscribed circle (by the way, it coincides with the height of the trapezoid):
Knowing the bases and the angle, the area of an isosceles trapezoid is calculated as follows:
S = a*b/sinα
(this and subsequent formulas are valid only for trapezoids with an inscribed circle).
Through the bases and the radius of the circle, the area is sought as follows:
If only the bases are known, then the area is calculated according to the formula:
Through the bases and the side line, the area of a trapezoid with an inscribed circle and through the bases and the midline - m is calculated as follows:
Area of a rectangular trapezoid
A trapezoid is called rectangular, in which one of the sides is perpendicular to the bases. In this case, the side length coincides with the height of the trapezoid.
A rectangular trapezoid is a square and a triangle. After finding the area of each of the figures, add up the results and get total area figures.
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 perpendicular side) are known, then the area is calculated by the formula:
S = (a + b) * h / 2
As h (height) can be the side with. Then the formula looks like this:
S = (a + b) * c / 2
- Another way to calculate area is to multiply the length of the midline by the height:
or by the length of the lateral perpendicular side:
- The next calculation method is through half the product of the diagonals and the sine of the angle between them:
S = ½ * d1 * d2 * sinα
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 midline.
Area of a curvilinear 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 x-axis and the straight lines x = a, x = b. In fact, two of its sides are parallel to each other (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:
How areas are calculated various kinds trapezium. But, in addition to the properties of the sides, trapezoids have the same properties of the angles. Like all existing quadrilaterals, the sum internal corners trapezium is 360 degrees. And the sum of the angles adjacent to the side is 180 degrees.
A trapezoid is a special kind of quadrilateral in which two opposite sides are parallel to each other and 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 for solving everyday or school problems.
Trapezoid geometry
A trapezoid (from the Greek "trapezion" - a 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 - the sides of the figure. The sides and their angles of inclination determine the type of trapezoid, which can be versatile, 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.
Trapeze in reality
AT Everyday life many real objects take on a trapezoidal shape. You can easily find trapeziums in the following areas of human activity:
- interior design and decor - sofas, countertops, walls, carpets, suspended ceilings;
- landscaping - lawn borders and artificial reservoirs, forms of decorative elements;
- fashion - the form of clothing, shoes and accessories;
- architecture - windows, walls, building foundations;
- production - various products and details.
With such a wide use of trapezoids, specialists often have to calculate the perimeter of a geometric figure.
Perimeter of a trapezoid
The perimeter of a figure is a numerical characteristic, which is calculated as the sum of the lengths of all sides of the n-gon. A trapezoid is a quadrilateral and in the general case all its sides have different lengths, so the perimeter is calculated by the formula:
P = a + b + c + d,
where a and c are the bases of the figure, b and d are its sides.
Even though we don't need to know the height when calculating the perimeter of a trapezoid, the calculator's code requires this variable to be entered. Since the height does not affect the calculation in any way, 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 an A-line scarf and you want to trim it with a fringe. You will need to know the perimeter of the scarf so that you do not buy extra material or go to the store twice. Let your isosceles scarf have following parameters: a = 120 cm, b = 60 cm, c = 100 cm, d = 60 cm. We drive this data into the online form and get the answer in the form:
Thus, the perimeter of the scarf is 340 cm, and this is the length of the fringe braid for its decoration.
slopes
For example, you decide to make slopes for non-standard plastic windows which are trapezoidal in shape. Such windows are widely used in the design of buildings, creating a composition of several shutters. Most often, such windows are made in the form of a rectangular trapezoid. Let's find out how much material is required to complete the slopes of such a window. 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 trapezoid window is 390 cm, and that's how much you need to buy plastic panels for the formation of slopes.
Conclusion
The trapezoid is a figure popular in everyday life, the definition of the parameters of which may be needed in the most unexpected situations. The calculation of perimeters by a trapezoid 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 tel.
