Area of an arbitrary trapezoid. Trapezoid perimeter calculator

The many-sided trapezoid... It can be arbitrary, isosceles or rectangular. And in each case you need to know how to find the area of a trapezoid. Of course, the easiest way is to remember the basic formulas. But sometimes it’s easier to use one that is derived taking into account all the features of a particular geometric figure.

A few words about the trapezoid and its elements

Any quadrilateral whose two sides are parallel can be called a trapezoid. In general, they are not equal and are called bases. The larger one is the lower one, and the other one is the upper one.

The other two sides turn out to be lateral. In an arbitrary trapezoid they have different lengths. If they are equal, then the figure becomes isosceles.

If suddenly the angle between any side and the base turns out to be equal to 90 degrees, then the trapezoid is rectangular.

All these features can help in solving the problem of how to find the area of a trapezoid.

Among the elements of the figure that may be indispensable in solving problems, we can highlight the following:

height, that is, a segment perpendicular to both bases;
the middle line, which has at its ends the midpoints of the lateral sides.

What formula can be used to calculate the area if the base and height are known?

This expression is given as a basic one because most often one can recognize these quantities even when they are not given explicitly. So, to understand how to find the area of a trapezoid, you will need to add both bases and divide them by two. Then multiply the resulting value by the height value.

If we denote the bases by the letters a 1 and a 2, the height by n, then the formula for the area will look like this:

S = ((a 1 + a 2)/2)*n.

The formula that calculates the area if its height and center line are given

If you look carefully at the previous formula, it is easy to notice that it clearly contains the value of the midline. Namely, the sum of the bases divided by two. Let the middle line be designated by the letter l, then the formula for the area becomes:

S = l * n.

Ability to find area using diagonals

This method will help if the angle formed by them is known. Suppose that the diagonals are designated by the letters d 1 and d 2, and the angles between them are α and β. Then the formula for how to find the area of a trapezoid will be written as follows:

S = ((d 1 * d 2)/2) * sin α.

You can easily replace α with β in this expression. The result will not change.

How to find out the area if all sides of the figure are known?

There are also situations when exactly the sides of this figure are known. This formula is cumbersome and difficult to remember. But it's possible. Let the sides have the designation: a 1 and a 2, the base a 1 is greater than a 2. Then the area formula will take the following form:

S = ((a 1 + a 2) / 2) * √ (in 1 2 - [(a 1 - a 2) 2 + in 1 2 - in 2 2) / (2 * (a 1 - a 2)) ] 2 ).

Methods for calculating the area of an isosceles trapezoid

The first is due to the fact that a circle can be inscribed in it. And, knowing its radius (it is denoted by the letter r), as well as the angle at the base - γ, you can use the following formula:

S = (4 * r 2) / sin γ.

The last general formula, which is based on knowledge of all sides of the figure, will be significantly simplified due to the fact that the sides have the same meaning:

S = ((a 1 + a 2) / 2) * √ (in 2 - [(a 1 - a 2) 2 / (2 * (a 1 - a 2))] 2 ).

Methods for calculating the area of a rectangular trapezoid

It is clear that any of the above is suitable for any figure. But sometimes it is useful to know about one feature of such a trapezoid. It lies in the fact that the difference between the squares of the lengths of the diagonals is equal to the difference made up of the squares of the bases.

Often the formulas for a trapezoid are forgotten, while the expressions for the areas of a rectangle and triangle are remembered. Then you can use a simple method. Divide the trapezoid into two shapes, if it is rectangular, or three. One will definitely be a rectangle, and the second, or the remaining two, will be triangles. After calculating the areas of these figures, all that remains is to add them up.

This is a fairly simple way to find the area of a rectangular trapezoid.

What if the coordinates of the vertices of the trapezoid are known?

In this case, you will need to use an expression that allows you to determine the distance between points. It can be applied three times: in order to find out both bases and one height. And then just apply the first formula, which is described a little higher.

To illustrate this method, the following example can be given. Given vertices with coordinates A(5; 7), B(8; 7), C(10; 1), D(1; 1). You need to find out the area of the figure.

Before finding the area of the trapezoid, you need to calculate the lengths of the bases from the coordinates. You will need the following formula:

length of the segment = √((difference of the first coordinates of the points) 2 + (difference of the second coordinates of the points) 2 ).

The upper base is designated AB, which means its length will be equal to √((8-5) 2 + (7-7) 2 ) = √9 = 3. The lower one is CD = √ ((10-1) 2 + (1-1 ) 2 ) = √81 = 9.

Now you need to draw the height from the top to the base. Let its beginning be at point A. The end of the segment will be on the lower base at the point with coordinates (5; 1), let this be point H. The length of the segment AN will be equal to √((5-5) 2 + (7-1) 2 ) = √36 = 6.

All that remains is to substitute the resulting values into the formula for the area of a trapezoid:

S = ((3 + 9) / 2) * 6 = 36.

The problem was solved without units of measurement, because the scale of the coordinate grid was not specified. It can be either a millimeter or a meter.

Examples of problems

No. 1. Condition. The angle between the diagonals of an arbitrary trapezoid is known; it is equal to 30 degrees. The smaller diagonal has a value of 3 dm, and the second is 2 times larger. It is necessary to calculate the area of the trapezoid.

Solution. First you need to find out the length of the second diagonal, because without this it will not be possible to calculate the answer. It is not difficult to calculate, 3 * 2 = 6 (dm).

Now you need to use the appropriate formula for area:

S = ((3 * 6) / 2) * sin 30º = 18/2 * ½ = 4.5 (dm 2). The problem is solved.

Answer: The area of the trapezoid is 4.5 dm2.

No. 2. Condition. In the trapezoid ABCD, the bases are the segments AD and BC. Point E is the middle of the SD side. From it a perpendicular is drawn to the straight line AB, the end of this segment is designated by the letter H. It is known that the lengths AB and EH are equal to 5 and 4 cm, respectively. It is necessary to calculate the area of the trapezoid.

Solution. First you need to make a drawing. Since the value of the perpendicular is less than the side to which it is drawn, the trapezoid will be slightly elongated upward. So EH will be inside the figure.

To clearly see the progress of solving the problem, you will need to perform additional construction. Namely, draw a straight line that will be parallel to side AB. The points of intersection of this line with AD are P, and with the continuation of BC are X. The resulting figure VHRA is a parallelogram. Moreover, its area is equal to the required one. This is due to the fact that the triangles that were obtained during additional construction are equal. This follows from the equality of the side and two angles adjacent to it, one vertical, the other lying crosswise.

You can find the area of a parallelogram using a formula that contains the product of the side and the height lowered onto it.

Thus, the area of the trapezoid is 5 * 4 = 20 cm 2.

Answer: S = 20 cm 2.

No. 3. Condition. The elements of an isosceles trapezoid have the following values: lower base - 14 cm, upper - 4 cm, acute angle - 45º. You need to calculate its area.

Solution. Let the smaller base be designated BC. The height drawn from point B will be called VH. Since the angle is 45º, triangle ABH will be rectangular and isosceles. So AN=VN. Moreover, AN is very easy to find. It is equal to half the difference in bases. That is (14 - 4) / 2 = 10 / 2 = 5 (cm).

The bases are known, the heights are calculated. You can use the first formula, which was discussed here for an arbitrary trapezoid.

S = ((14 + 4) / 2) * 5 = 18/2 * 5 = 9 * 5 = 45 (cm 2).

Answer: The required area is 45 cm 2.

No. 4. Condition. There is an arbitrary trapezoid ABCD. Points O and E are taken on its lateral sides, so that OE is parallel to the base of AD. The area of the AOED trapezoid is five times larger than that of the OVSE. Calculate the OE value if the lengths of the bases are known.

Solution. You will need to draw two parallel lines AB: the first through point C, its intersection with OE is point T; the second through E and the point of intersection with AD will be M.

Let the unknown OE=x. The height of the smaller trapezoid OVSE is n 1, the larger AOED is n 2.

Since the areas of these two trapezoids are related as 1 to 5, we can write the following equality:

(x + a 2) * n 1 = 1/5 (x + a 1) * n 2

n 1 / n 2 = (x + a 1) / (5 (x + a 2)).

The heights and sides of the triangles are proportional by construction. Therefore, we can write one more equality:

n 1 / n 2 = (x - a 2) / (a 1 - x).

In the last two entries on the left side there are equal values, which means we can write that (x + a 1) / (5(x + a 2)) is equal to (x - a 2) / (a 1 - x).

A number of transformations are required here. First multiply crosswise. Parentheses will appear to indicate the difference of squares, after applying this formula you will get a short equation.

In it you need to open the brackets and move all the terms with the unknown “x” to the left, and then extract the square root.

Answer: x = √ ((a 1 2 + 5 a 2 2) / 6).

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.

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 solution is selected depending on the given 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 in various 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 certain 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.

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 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))

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) 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 x-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.

Area of a trapezoid. Greetings! In this publication we will look at this formula. Why is she exactly like this and how to understand her. If there is understanding, then you don’t need to teach it. If you just want to look at this formula and urgently, then you can immediately scroll down the page))

Now in detail and in order.

A trapezoid is a quadrilateral, two sides of this quadrilateral are parallel, the other two are not. Those that are not parallel are the bases of the trapezoid. The other two are called sides.

If the sides are equal, then the trapezoid is called isosceles. If one of the sides is perpendicular to the bases, then such a trapezoid is called rectangular.

In its classic form, a trapezoid is depicted as follows - the larger base is at the bottom, respectively, the smaller one is at the top. But no one forbids depicting her and vice versa. Here are the sketches:

Next important concept.
The midline of a trapezoid is a segment that connects the midpoints of the sides. The middle line is parallel to the bases of the trapezoid and equal to their half-sum.

Now let's delve deeper. Why is this so?

Consider a trapezoid with bases a and b and with the middle line l, and let’s perform some additional constructions: draw straight lines through the bases, and perpendiculars through the ends of the midline until they intersect with the bases:

*Letter designations for vertices and other points are not included intentionally to avoid unnecessary designations.

Look, triangles 1 and 2 are equal according to the second sign of equality of triangles, triangles 3 and 4 are the same. From the equality of triangles follows the equality of the elements, namely the legs (they are indicated in blue and red, respectively).

Now attention! If we mentally “cut off” the blue and red segments from the lower base, then we will be left with a segment (this is the side of the rectangle) equal to the middle line. Next, if we “glue” the cut blue and red segments to the upper base of the trapezoid, then we will also get a segment (this is also the side of the rectangle) equal to the midline of the trapezoid.

Got it? It turns out that the sum of the bases will be equal to the two middle lines of the trapezoid:

View another explanation

Let's do the following - construct a straight line passing through the lower base of the trapezoid and a straight line that will pass through points A and B:

We get triangles 1 and 2, they are equal along the side and adjacent angles (the second sign of equality of triangles). This means that the resulting segment (in the sketch it is indicated in blue) is equal to the upper base of the trapezoid.

Now consider the triangle:

*The midline of this trapezoid and the midline of the triangle coincide.

It is known that a triangle is equal to half of the base parallel to it, that is:

Okay, we figured it out. Now about the area of the trapezoid.

Trapezoid area formula:

They say: the area of a trapezoid is equal to the product of half the sum of its bases and height.

That is, it turns out that it is equal to the product of the center line and the height:

You've probably already noticed that this is obvious. Geometrically, this can be expressed this way: if we mentally cut off triangles 2 and 4 from the trapezoid and place them on triangles 1 and 3, respectively:

Then we will get a rectangle with an area equal to the area of our trapezoid. The area of this rectangle will be equal to the product of the center line and the height, that is, we can write: