The area of ​​a trapezoid is equal to the formula. Trapezoid

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 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:

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

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That's all. Good luck to you!

Sincerely, Alexander.

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

A trapezoid is a geometric figure formed by four intersecting segments, two of which are parallel to each other and are called the bases of the trapezoid. The other two segments are called the sides of the trapezoid. In addition, we will need another definition later. This is the middle line of the trapezoid, which is a segment connecting the midpoints of the sides and the height of the trapezoid, which is equal to the distance between the bases.
Like triangles, trapezoids have special types in the form of an isosceles (equal-sided) trapezoid, in which the lengths of the sides are the same, and a rectangular trapezoid, in which one of the sides forms a right angle with the bases.

Trapezes have some interesting properties:

1. The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.
   
2. Isosceles trapezoids have equal sides and the angles they form with the bases.
   
3. The midpoints of the diagonals of a trapezoid and the point of intersection of its diagonals are on the same straight line.
   
4. 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
   
5. 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.
   
6. An isosceles trapezoid can be described by a circle. And vice versa. If a trapezoid fits into a circle, then it is isosceles.
   
7. 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 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.

Before finding the area of ​​a trapezoid, it is necessary to determine the known elements of the trapezoid. A trapezoid is a geometric object, namely a quadrilateral that has two parallel sides (two bases). The other two sides are lateral. If these two sides of the quadrilateral are also parallel, then it will no longer be a trapezoid, but a parallelogram. If at least one angle of a trapezoid is 90 degrees, then such a trapezoid is called rectangular. We'll look at how to find the area of ​​a rectangular trapezoid later. There is also an isosceles trapezoid, the name of which speaks for itself: the sides of such a trapezoid are equal. The distance between the bases of a trapezoid is called the height, and height is very often used to find area. The midline of a trapezoid is a segment that connects the midpoints of the sides.

Basic formulas for finding the area of ​​a trapezoid

• S= h*(a+b)/2
  Where h is the height of the trapezoid, a, b are the bases. The most commonly used formula for finding the area of ​​a trapezoid is half the sum of the bases multiplied by the height.
• S = m*h
  Where m is the midline of the trapezoid, h is the height. The area of ​​a trapezoid is also equal to the product of the midline of the trapezoid and its height.
• S=1/2*d1*d2*sin(d1^d2)
  Where d1, d2 are the diagonals of the trapezoid, sin(d1^d2) is the sine of the angle between the diagonals of the trapezoid.

There are also various formulas derived from the basic ones, as well as a formula for calculating the area of ​​a trapezoid when all its sides are known. However, this formula is quite cumbersome and is rarely used, because, knowing all the sides of the trapezoid, you can simply determine the height or its midline. You can also inscribe a circle in an isosceles trapezoid. In this case, the area of ​​the trapezoid will be calculated using the formula: 8 * radius of the circle squared.

How to find the area of ​​a rectangular trapezoid

As mentioned earlier, a trapezoid is called rectangular if it has at least one right angle. Finding the area of ​​such a trapezoid is very simple. Basically, to find the area of ​​a rectangular trapezoid, the same formulas are used as for a regular trapezoid. However, it is worth remembering that one of the sides of such a trapezoid will be the height. Also, often solving problems of finding the area of ​​a rectangular trapezoid comes down to finding the area of ​​the rectangle and triangle formed by the omitted height. Such tasks are quite simple.

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 designate the bases as a 1 and a 2, and the height as 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 probably. 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.

Sample 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. A perpendicular to straight line AB is drawn from it, 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 base - 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 - 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 that 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).

Instructions

To make both methods more understandable, we can give a couple of examples.

Example 1: the length of the midline of the trapezoid is 10 cm, its area is 100 cm². To find the height of this trapezoid, you need to do:

h = 100/10 = 10 cm

Answer: the height of this trapezoid is 10 cm

Example 2: the area of ​​the trapezoid is 100 cm², the lengths of the bases are 8 cm and 12 cm. To find the height of this trapezoid, you need to perform the following action:

h = (2*100)/(8+12) = 200/20 = 10 cm

Answer: the height of this trapezoid is 20 cm

note

There are several types of trapezoids:
An isosceles trapezoid is a trapezoid in which the sides are equal to each other.
A right-angled trapezoid is a trapezoid with one of its interior angles measuring 90 degrees.
It is worth noting that in a rectangular trapezoid, the height coincides with the length of the side at a right angle.
You can draw a circle around a trapezoid, or fit it inside a given figure. You can inscribe a circle only if the sum of its bases is equal to the sum of its opposite sides. A circle can only be described around an isosceles trapezoid.

Helpful advice

A parallelogram is a special case of a trapezoid, because the definition of a trapezoid does not in any way contradict the definition of a parallelogram. A parallelogram is a quadrilateral whose opposite sides are parallel to each other. For a trapezoid, the definition refers only to a pair of its sides. Therefore, any parallelogram is also a trapezoid. The reverse statement is not true.

Sources:

• how to find the area of ​​a trapezoid formula

Tip 2: How to find the height of a trapezoid if the area is known

A trapezoid is a quadrilateral in which two of its four sides are parallel to each other. The parallel sides are the bases of the given one, the other two are the lateral sides of the given one. trapezoids. Find height trapezoids, if known square, it will be very easy.

Instructions

You need to figure out how to calculate square original trapezoids. There are several formulas for this, depending on the initial data: S = ((a+b)*h)/2, where a and b are bases trapezoids, and h is its height (Height trapezoids- perpendicular, lowered from one base trapezoids to another);
S = m*h, where m is line trapezoids(The middle line is a segment with bases trapezoids and connecting the midpoints of its sides).

To make it clearer, similar problems can be considered: Example 1: Given a trapezoid with square 68 cm², the middle line of which is 8 cm, you need to find height given trapezoids. In order to solve this problem, you need to use the previously derived formula:
h = 68/8 = 8.5 cm Answer: height of this trapezoids is 8.5 cmExample 2: Let y trapezoids square equals 120 cm², the length of the bases of this trapezoids 8 cm and 12 cm respectively, you need to find height this trapezoids. To do this, you need to apply one of the derived formulas:
h = (2*120)/(8+12) = 240/20 = 12 cmAnswer: given height trapezoids equal to 12 cm

Video on the topic

note

Any trapezoid has a number of properties:

The midline of a trapezoid is equal to half the sum of its bases;

The segment that connects the diagonals of a trapezoid is equal to half the difference of its bases;

If a straight line is drawn through the midpoints of the bases, then it will intersect the point of intersection of the diagonals of the trapezoid;

A circle can be inscribed in a trapezoid if the sum of the bases of the trapezoid is equal to the sum of its sides.

Use these properties when solving problems.

Tip 3: How to find the area of ​​a trapezoid if the bases are known

By geometric definition, a trapezoid is a quadrilateral with only one pair of sides parallel. These sides are hers reasons. Distance between reasons called height trapezoids. Find square trapezoids possible using geometric formulas.

Instructions

Measure the bases and trapezoids ABCD. Usually they are given in tasks. Let in this example problem the base AD (a) trapezoids will be equal to 10 cm, base BC (b) - 6 cm, height trapezoids BK (h) - 8 cm. Use geometric to find area trapezoids, if the lengths of its bases and heights are known - S= 1/2 (a+b)*h, where: - a - the size of the base AD trapezoids ABCD, - b - the value of the base BC, - h - the value of the height BK.