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How to find the area of a trapezoid if. How to find the area of an isosceles trapezoid

AND . Now we can begin to consider the question of how to find the area of a trapezoid. This task in everyday life occurs very rarely, but sometimes it turns out to be necessary, for example, to find the area of a room in the form of a trapezoid, which is increasingly used in the construction of modern apartments, or in renovation design projects.

Trapeze is geometric figure, formed by four intersecting segments, two of which are parallel to each other and are called the bases of a trapezoid. The other two segments are called the sides of the trapezoid. In addition, we will need another definition later 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.
A segment passing through the midpoints of the bases isosceles trapezium 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 one more simple formula to calculate 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 middle 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 \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 trapezium. 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 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 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 of the interior angles of a trapezoid is 360 degrees. And the sum of the angles adjacent to the side is 180 degrees.

The area of the trapezoid. Greetings! In this publication, we will consider this formula. Why is it the way it is and how can you understand it? If there is an understanding, then you do not need to learn it. If you just want to see this formula and what is urgent, 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 trapezium. 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 the classical form, the 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 it and vice versa. Here are the sketches:

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

Now let's delve deeper. Why exactly?

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

*Letter designations of vertices and other points are not entered 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 respectively in blue and red).

Now attention! If we mentally “cut off” the blue and red segments from the lower base, then we will have a segment (this is the side of the rectangle) equal to the midline. Further, if we “glue” the cut off 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 medians of the trapezoid:

See another explanation

Let's do the following - build 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 in side and adjacent angles (the second sign of equality of triangles). This means that the resulting segment (in the sketch it is marked in blue) is equal to the upper base of the trapezoid.

Now consider a triangle:

*The median line of this trapezoid and the median line of the triangle coincide.

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

Okay, got it. Now about the area of the trapezoid.

Trapezium 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 midline and height:

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

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

But the point here is not in writing, of course, but in understanding.

Download (view) the material of the article in *pdf format

That's all. Good luck to you!

Sincerely, Alexander.

The section contains problems in geometry (section planimetry) about trapezoids. If you did not find a solution to the problem - write about it on the forum. The course will be updated for sure.

Trapeze. Definition, formulas and properties

A trapezium (from other Greek τραπέζιον - “table”; τράπεζα - “table, food”) is a quadrilateral with exactly one pair of opposite sides parallel.

A trapezoid is a quadrilateral with two opposite sides parallel.

Note. In this case, the parallelogram is a special case of a trapezoid.

The parallel opposite sides are called the bases of the trapezoid, and the other two are called the sides.

Trapezes are:

- versatile ;

- isosceles;

- rectangular

.
red and brown flowers the lateral sides are indicated, green and blue are the bases of the trapezoid.

A - isosceles (isosceles, isosceles) trapezoid
B - rectangular trapezoid
C - versatile trapezoid

A versatile trapezoid has all sides of different lengths, and the bases are parallel.

The sides are equal and the bases are parallel.

They are parallel at the base, one side is perpendicular to the bases, and the second side is inclined towards the bases.

Trapezoid Properties

Median line of the trapezoid parallel to the bases and equal to half their sum
A line segment connecting the midpoints of the diagonals, is equal to half the difference of the bases and lies on the midline. Its length
Parallel lines intersecting the sides of any angle of the trapezoid cut off proportional segments from the sides of the angle (see Thales' theorem)
Intersection point of the diagonals of a trapezoid, the point of intersection of the extensions of its lateral sides and the midpoints of the bases lie on one straight line (see also the properties of a quadrilateral)
Triangles on bases trapezoids whose vertices are the intersection point of their diagonals are similar. The ratio of the areas of such triangles is equal to the square of the ratio of the bases of the trapezoid
Triangles on the sides trapeziums whose vertices are the point of intersection of its diagonals are equal in area (equal in area)
into a trapezoid you can inscribe a circle if the sum of the lengths of the bases of a trapezoid is equal to the sum of the lengths of its sides. The median line in this case is equal to the sum of the sides divided by 2 (since the median line of the trapezoid is equal to half the sum of the bases)
A segment parallel to the bases and passing through the intersection point of the diagonals, is divided by the latter in half and is equal to twice the product of the bases divided by their sum 2ab / (a + b) (Burakov's formula)

Trapeze angles

Trapeze angles are sharp, straight and blunt.
There are only two right angles.

A rectangular trapezoid has two right angles, and the other two are acute and blunt. Other types of trapeziums have: two sharp corners and two dumb ones.

The obtuse angles of a trapezoid belong to the smallest along the length of the base, and sharp - more basis.

Any trapezoid can be considered like a truncated triangle, whose section line is parallel to the base of the triangle.
Important. Please note that in this way (by additional construction of a trapezoid to a triangle) some problems about a trapezoid can be solved and some theorems can be proved.

How to find the sides and diagonals of a trapezoid

Finding the sides and diagonals of a trapezoid is done using the formulas that are given below:

In these formulas, the notation is used, as in the figure.

a - the smallest of the bases of the trapezoid
b - the largest of the bases of the trapezoid
c,d - sides
h 1 h 2 - diagonals

The sum of the squares of the diagonals of a trapezoid is equal to twice the product of the bases of the trapezoid plus the sum of the squares of the sides (Formula 2)

Trapeze is called a quadrilateral only two sides are parallel to each other.

They are called the bases of the figure, the rest - the sides. A parallelogram is considered a special case of a figure. There is also a curvilinear trapezoid, which includes a function graph. The 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 height and midline. middle line- this is a line connecting the midpoints of the sides. Height trapezium is held at right angles from top corner to the base.
The area of a trapezoid through the height is equal to the product of half the sum of the lengths of the bases, multiplied by the height:

If the median line is known according to the conditions, then this formula is greatly 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 through these data:

Suppose a trapezoid is given with bases a = 3 cm, b = 7 cm and sides c = 5 cm, d = 4 cm. Find the area of the figure:

Area of an isosceles trapezoid

A separate case is an isosceles or, as it is also called, an isosceles trapezoid.
A special case is also finding the area of an isosceles (isosceles) trapezoid. Formula derived different ways- through the diagonals, through the angles adjacent to the base and the radius of the inscribed circle.
If the length of the diagonals is specified by 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 curvilinear trapezoid

A separate case is curvilinear trapezoid. It is located on the coordinate axis and is limited to a 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 curvilinear trapezoid.
The formula is written like this:

Consider an example of calculating the area of a curvilinear trapezoid. The formula requires certain knowledge to work with definite integrals. First, let's analyze 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 curvilinear trapezoid, function limited. Function
We need to find the area of the selected figure, which is a curvilinear trapezoid bounded on top by a graph, on the right is a straight line x = (-8), on the left is a straight line x = (-10) and the axis OX is below.
We will calculate the area of this figure using the formula:

We are given a function by the conditions of the problem. Using it, we will find the values of the antiderivative at each of our points:

Now
Answer: the area of a given curvilinear trapezoid is 4.