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 special case trapezoids. 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 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 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 the perpendicular side side) are known, then the area is calculated using the formula:

S = (a + b) * h / 2

The side side c can act as h (height). Then the formula looks like this:

S = (a + b) * c / 2

• Another way to calculate area is to multiply the length of the center line by the height:

or by the length of the lateral perpendicular side:

• The next way to calculate is through half the product of the diagonals and the sine of the angle between them:

S = ½ * d1 * d2 * sin α

If the diagonals are perpendicular, then the formula simplifies to:

S = ½ * d1 * d2

• Another way to calculate is through the semi-perimeter (the sum of the lengths of two opposite sides) and the radius of the inscribed circle.

This formula is valid for bases. If we take the lengths of the sides, then one of them will be equal to twice the radius. The formula will look like this:

S = (2r + c) * r

• If a circle is inscribed in a trapezoid, then the area is calculated in the same way:

where m is the length of the center line.

Area of ​​a curved trapezoid

A curvilinear trapezoid is a flat figure bounded by the graph of a non-negative continuous function y = f(x), defined on the segment, the 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.

Square curved trapezoid search through the integral using the Newton-Leibniz formula:

This is how areas are calculated various types trapezoid. 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.

A trapezoid is a relief quadrilateral in which two opposite sides are parallel and the other two are non-parallel. If all opposite sides of a quadrilateral are parallel in pairs, then it is a parallelogram.

You will need

• – all sides of the trapezoid (AB, BC, CD, DA).

Instructions

1. Non-parallel sides trapezoids are called lateral sides, and parallel sides are called bases. The line between the bases, perpendicular to them - height trapezoids. If lateral sides trapezoids are equal, then it is called isosceles. First, let's look at the solution for trapezoids, which is not isosceles.

2. Draw line segment BE from point B to the lower base AD parallel to the side trapezoids CD. Because BE and CD are parallel and drawn between parallel bases trapezoids BC and DA, then BCDE is a parallelogram, and its opposite sides BE and CD are equal. BE=CD.

3. Look at the triangle ABE. Calculate side AE. AE=AD-ED. Reasons trapezoids BC and AD are known, and in a parallelogram BCDE are opposite sides ED and BC are equal. ED=BC, so AE=AD-BC.

4. Now find out the area of ​​triangle ABE using Heron's formula by calculating the semi-perimeter. S=root(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is the semi-perimeter of triangle ABE. p=1/2*(AB+BE+AE). To calculate the area, you know all the necessary data: AB, BE=CD, AE=AD-BC.

6. Express from this formula the height of the triangle, which is also the height trapezoids. BH=2*S/AE. Calculate it.

7. If the trapezoid is isosceles, the solution can be executed differently. Look at the triangle ABH. It is rectangular because one of the corners, BHA, is right.

8. Draw height CF from vertex C.

9. Study the HBCF figure. HBCF rectangle, because there are two of it sides are heights, and the other two are bases trapezoids, that is, the angles are right, and the opposite sides parallel. This means that BC=HF.

10. Look at the right triangles ABH and FCD. The angles at heights BHA and CFD are right, and the angles at lateral sides x BAH and CDF are equal because the trapezoid ABCD is isosceles, which means the triangles are similar. Because the heights BH and CF are equal or lateral sides isosceles trapezoids AB and CD are congruent, then similar triangles are congruent. So they sides AH and FD are also equal.

11. Discover AH. AH+FD=AD-HF. Because from a parallelogram HF=BC, and from triangles AH=FD, then AH=(AD-BC)*1/2.

Trapezoid – geometric figure, which is a quadrilateral in which two sides, called bases, are parallel, and the other two are not parallel. They are called sides trapezoids. The segment drawn through the midpoints of the lateral sides is called the midline trapezoids. A trapezoid can have different side lengths or identical ones, in which case it is called isosceles. If one of the sides is perpendicular to the base, then the trapezoid will be rectangular. But it is much more practical to know how to detect square trapezoids .

You will need

• Ruler with millimeter divisions

Instructions

1. Measure all sides trapezoids: AB, BC, CD and DA. Record your measurements.

2. On segment AB, mark the middle - point K. On segment DA, mark point L, which is also in the middle of segment AD. Combine points K and L, the resulting segment KL will be the middle line trapezoids ABCD. Measure the segment KL.

3. From the top trapezoids– toss C, lower the perpendicular to its base AD on the segment CE. It will be the height trapezoids ABCD. Measure the segment CE.

4. Let us call the segment KL the letter m, and the segment CE the letter h, then square S trapezoids ABCD is calculated using the formula: S=m*h, where m is the middle line trapezoids ABCD, h – height trapezoids ABCD.

5. There is another formula that allows you to calculate square trapezoids ABCD. Bottom base trapezoids– Let’s call AD the letter b, and the upper base BC the letter a. The area is determined by the formula S=1/2*(a+b)*h, where a and b are the bases trapezoids, h – height trapezoids .

Video on the topic

Tip 3: 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. Parallel sides are the bases of this trapezoids, the other two are the lateral sides of this trapezoids. Discover height trapezoids, if you know its area, it will be very easy.

Instructions

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

2. Now, knowing the formulas for calculating area trapezoids, it is allowed to derive new ones from them to find the height trapezoids:h = (2*S)/(a+b);h = S/m.

3. In order to make it clearer how to solve similar problems, you can look at examples: Example 1: Given a trapezoid whose area is 68 cm?, 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: the height of this trapezoids is 8.5 cmExample 2: Let y trapezoids area is 120 cm?, the length of the bases is given trapezoids are equal to 8 cm and 12 cm respectively, it is required to detect height this trapezoids. To do this, you need to apply one of the derived formulas: h = (2*120)/(8+12) = 240/20 = 12 cmAnswer: height of the given trapezoids equal to 12 cm

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Pay attention!
Any trapezoid has a number of properties: - the middle line of a trapezoid is equal to half the sum of its bases; - the segment that connects the diagonals of the 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; - You can inscribe a circle into a trapezoid if the sum of the bases of a given trapezoid is equal to the sum of its sides. Use these properties when solving problems.

Tip 4: How to find the height of a triangle given the coordinates of the points

The height in a triangle is the straight line segment connecting the vertex of the figure to the opposite side. This segment must certainly be perpendicular to the side; therefore, from any vertex it is allowed to draw only one height. Because there are three vertices in this figure, there are the same number of heights. If a triangle is given by the coordinates of its vertices, the length of each of the heights can be calculated, say, using the formula for finding the area and calculating the lengths of the sides.

Instructions

1. Proceed in your calculations from the fact that the area triangle is equal to half the product of the length of each of its sides by the length of the height lowered onto this side. From this definition it follows that to find the height you need to know the area of ​​the figure and the length of the side.

2. Start by calculating the lengths of the sides triangle. Designate the coordinates of the vertices of the figure as follows: A(X?,Y?,Z?), B(X?,Y?,Z?) and C(X?,Y?,Z?). Then you can calculate the length of side AB using the formula AB = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?). For the other 2 sides, these formulas will look like this: BC = ?((X?-X?)? + (Y?-Y?)? + (Z?-Z?)?) and AC = ?((X ?-X?)? + (Y?-Y?)? + (Z?-Z?)?). Let's say for triangle with coordinates A(3,5,7), B(16,14,19) and C(1,2,13) ​​the length of side AB will be?((3-16)? + (5-14)? + (7 -19)?) = ?(-13? + (-9?) + (-12?)) = ?(169 + 81 + 144) = ?394 ? 19.85. The lengths of the sides BC and AC, calculated by the same method, will be equal?(15? + 12? + 6?) = ?405? 20.12 and?(2? + 3? + (-6?)) =?49 = 7.

3. Knowing the lengths of 3 sides obtained in the previous step is enough to calculate the area triangle(S) according to Heron’s formula: S = ? * ?((AB+BC+CA) * (BC+CA-AB) * (AB+CA-BC) * (AB+BC-CA)). Let's say, after substituting into this formula the values ​​​​obtained from the coordinates triangle-example from the previous step, this formula will give the following value: S = ?*?((19.85+20.12+7) * (20.12+7-19.85) * (19.85+7-20 .12) * (19.85+20.12-7)) = ?*?(46.97 * 7.27 * 6.73 * 32.97) ? ?*?75768.55 ? ?*275.26 = 68.815.

4. Based on area triangle, calculated in the previous step, and the lengths of the sides obtained in the second step, calculate the heights for each of the sides. Because the area is equal to half the product of the height and the length of the side to which it is drawn, to find the height, divide the doubled area by the length of the required side: H = 2*S/a. For the example used above, the height lowered to side AB will be 2*68.815/16.09? 8.55, the height to the BC side will have a length of 2*68.815/20.12? 6.84, and for the AC side this value will be equal to 2*68.815/7? 19.66.

There are many ways to find the area of ​​a trapezoid. Usually a math tutor knows several methods of calculating it, let’s look at them in more detail:
1) , where AD and BC are the bases, and BH is the height of the trapezoid. Proof: draw the diagonal BD and express the areas of triangles ABD and CDB through the half product of their bases and heights:

, where DP is the external height in

Let us add these equalities term by term and taking into account that the heights BH and DP are equal, we obtain:

Let's put it out of brackets

Q.E.D.

Corollary to the formula for the area of ​​a trapezoid:
Since the half-sum of the bases is equal to MN - the midline of the trapezoid, then

2) Application general formula area of ​​a quadrilateral.
The area of ​​a quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them
To prove it, it is enough to divide the trapezoid into 4 triangles, express the area of ​​each through “half the product of the diagonals and the sine of the angle between them” (taken as the angle, add the resulting expressions, take them out of the bracket and factor this bracket using the grouping method to obtain its equality to the expression. Hence

3) Diagonal shift method
This is my name. A math tutor will not come across such a heading in school textbooks. A description of the technique can only be found in additional textbooks as an example of solving a problem. I note that most of the interesting and useful facts mathematics tutors reveal planimetry to students in the process of performing practical work. This is extremely suboptimal, because the student needs to isolate them into separate theorems and call them “ big names" One of these is “diagonal shift”. About what we're talking about?Let us draw a line parallel to AC through vertex B until it intersects with the lower base at point E. In this case, the quadrilateral EBCA will be a parallelogram (by definition) and therefore BC=EA and EB=AC. The first equality is important to us now. We have:

Note that the triangle BED, whose area is equal to the area of ​​the trapezoid, has several more remarkable properties:
1) Its area is equal to the area of ​​the trapezoid
2) Its isosceles occurs simultaneously with the isosceles of the trapezoid itself
3) Its upper angle at vertex B is equal to the angle between the diagonals of the trapezoid (which is very often used in problems)
4) Its median BK is equal to the distance QS between the midpoints of the bases of the trapezoid. I recently encountered the use of this property when preparing a student for Mechanics and Mathematics at Moscow State University using Tkachuk’s textbook, 1973 version (the problem is given at the bottom of the page).

Special techniques for a math tutor.

Sometimes I propose problems using a very tricky way of finding the area of ​​a trapezoid. I classify it as a special technique because in practice the tutor uses them extremely rarely. If you need preparation for the Unified State Exam in mathematics only in Part B, you don’t have to read about them. For others, I'll tell you further. It turns out that the area of ​​the trapezoid is twice more area a triangle with vertices at the ends of one side and the middle of the other, that is, the ABS triangle in the figure:
Proof: draw the heights SM and SN in triangles BCS and ADS and express the sum of the areas of these triangles:

Since point S is the midpoint of CD, then (prove it yourself). Let’s find the sum of the areas of the triangles:

Since this sum turned out to be equal to half the area of ​​the trapezoid, then its second half. Etc.

I would include in the tutor’s collection of special techniques the form of calculating the area of ​​an isosceles trapezoid along its sides: where p is the semi-perimeter of the trapezoid. I won't give proof. Otherwise, your math tutor will be left without a job :). Come to class!

Problems on the area of ​​a trapezoid:

Math tutor's note: The list below is not a methodological accompaniment to the topic, it is only a small selection of interesting tasks based on the techniques discussed above.

1) The lower base of an isosceles trapezoid is 13, and the upper is 5. Find the area of ​​the trapezoid if its diagonal is perpendicular to the side.
2) Find the area of ​​a trapezoid if its bases are 2cm and 5cm, and its sides are 2cm and 3cm.
3) In an isosceles trapezoid, the larger base is 11, the side is 5, and the diagonal is Find the area of ​​the trapezoid.
4) The diagonal of an isosceles trapezoid is 5 and the midline is 4. Find the area.
5) In an isosceles trapezoid, the bases are 12 and 20, and the diagonals are mutually perpendicular. Calculate the area of ​​a trapezoid
6) The diagonal of an isosceles trapezoid makes an angle with its lower base. Find the area of ​​the trapezoid if its height is 6 cm.
7) The area of ​​the trapezoid is 20, and one of its sides is 4 cm. Find the distance to it from the middle of the opposite side.
8) The diagonal of an isosceles trapezoid divides it into triangles with areas of 6 and 14. Find the height if the lateral side is 4.
9) In a trapezoid, the diagonals are equal to 3 and 5, and the segment connecting the midpoints of the bases is equal to 2. Find the area of ​​the trapezoid (Mekhmat MSU, 1970).

I chose not the most difficult problems (don’t be afraid of mechanical engineering!) with the expectation that I would be able to solve them independently. Decide for your health! If you need preparation for the Unified State Exam in mathematics, then without participation in this process, formulas for the area of ​​a trapezoid may arise serious problems even with problem B6 and even more so with C4. Do not start the topic and in case of any difficulties, ask for help. A math tutor is always happy to help you.

Kolpakov A.N.
Mathematics tutor in Moscow, preparation for the Unified State Exam in Strogino.

A trapezoid is a quadrilateral whose two sides are parallel (these are the bases of the trapezoid, indicated in the figure a and b), and the other two are not (in the figure AD and CB). The height of a trapezoid is a segment h drawn perpendicular to the bases.

How to find the height of a trapezoid given the known values ​​of the area of ​​the trapezoid and the lengths of the bases?

To calculate the area S of the trapezoid ABCD, we use the formula:

S = ((a+b) × h)/2.

Here segments a and b are the bases of the trapezoid, h is the height of the trapezoid.

Transforming this formula, we can write:

Using this formula, we obtain the value of h if the area S and the lengths of the bases a and b are known.

Example

If it is known that the area of ​​the trapezoid S is 50 cm², the length of the base a is 4 cm, and the length of the base b is 6 cm, then to find the height h, we use the formula:

We substitute known quantities into the formula.

h = (2 × 50)/(4+6) = 100/10 = 10 cm

Answer: The height of the trapezoid is 10 cm.

How to find the height of a trapezoid if the area of ​​the trapezoid and the length of the midline are given?

Let's use the formula for calculating the area of ​​a trapezoid:

Here m is the middle line, h is the height of the trapezoid.

If the question arises, how to find the height of a trapezoid, the formula is:

h = S/m will be the answer.

Thus, we can find the height of the trapezoid h, given the known values ​​of the area S and the midline segment m.

Example

The length of the midline of the trapezoid m, which is 20 cm, and the area S, which is 200 cm², are known. Let's find the value of the height of the trapezoid h.

Substituting the values ​​of S and m, we get:

h = 200/20 = 10 cm

Answer: the height of the trapezoid is 10 cm

How to find the height of a rectangular trapezoid?

If a trapezoid is a quadrilateral, with two parallel sides (bases) of the trapezoid. Then a diagonal is a segment that connects two opposite vertices of the corners of a trapezoid (segment AC in the figure). If the trapezoid is rectangular, using the diagonal, we find the height of the trapezoid h.

A rectangular trapezoid is a trapezoid where one of the sides is perpendicular to the bases. In this case, its length (AD) coincides with the height h.

So, consider a rectangular trapezoid ABCD, where AD is the height, DC is the base, AC is the diagonal. Let's use the Pythagorean theorem. Hypotenuse square AC right triangle ADC equal to the sum the squares of its legs AB and BC.

Then we can write:

AC² = AD² + DC².

AD is the leg of the triangle, the lateral side of the trapezoid and, at the same time, its height. After all, the segment AD is perpendicular to the bases. Its length will be:

AD = √(AC² - DC²)

So, we have a formula for calculating the height of a trapezoid h = AD

Example

If the length of the base of a rectangular trapezoid (DC) is 14 cm, and the diagonal (AC) is 15 cm, we use the Pythagorean theorem to obtain the value of the height (AD - side).

Let x be the unknown leg of a right triangle (AD), then

AC² = AD² + DC² can be written

15² = 14² + x²,

x = √(15²-14²) = √(225-196) = √29 cm

Answer: the height of a rectangular trapezoid (AB) will be √29 cm, which is approximately 5.385 cm

How to find the height of an isosceles trapezoid?

An isosceles trapezoid is a trapezoid whose side lengths are equal to each other. The straight line drawn through the midpoints of the bases of such a trapezoid will be the axis of symmetry. A special case is a trapezoid, the diagonals of which are perpendicular to each other, then the height h will be equal to half the sum of the bases.

Let's consider the case if the diagonals are not perpendicular to each other. In an equilateral (isosceles) trapezoid, the angles at the bases are equal and the lengths of the diagonals are equal. It is also known that all vertices of an isosceles trapezoid touch the line of a circle drawn around this trapezoid.

Let's look at the drawing. ABCD is an isosceles trapezoid. It is known that the bases of the trapezoid are parallel, which means that BC = b is parallel to AD = a, side AB = CD = c, which means that the angles at the bases are correspondingly equal, we can write the angle BAQ = CDS = α, and the angle ABC = BCD = β. Thus, we conclude that triangle ABQ is equal to triangle SCD, which means the segment

AQ = SD = (AD - BC)/2 = (a - b)/2.

Having, according to the conditions of the problem, the values ​​of the bases a and b, and the length of the side side c, we find the height of the trapezoid h, equal to the segment BQ.

Consider right triangle ABQ. VO is the height of the trapezoid, perpendicular to the base AD, and therefore to the segment AQ. We find side AQ of triangle ABQ using the formula we derived earlier:

Having the values ​​of two legs of a right triangle, we find the hypotenuse BQ = h. We use the Pythagorean theorem.

AB²= AQ² + BQ²

Let's substitute these tasks:

c² = AQ² + h².

We obtain a formula for finding the height of an isosceles trapezoid:

h = √(c²-AQ²).

Example

Given an isosceles trapezoid ABCD, where base AD = a = 10cm, base BC = b = 4cm, and side AB = c = 12cm. Under such conditions, let's look at an example of how to find the height of a trapezoid, an isosceles trapezoid ABCD.

Let's find side AQ of triangle ABQ by substituting the known data:

AQ = (a - b)/2 = (10-4)/2=3cm.

Now let's substitute the values ​​of the sides of the triangle into the formula of the Pythagorean theorem.

h = √(c²- AQ²) = √(12²- 3²) = √135 = 11.6 cm.

Answer. The height h of the isosceles trapezoid ABCD is 11.6 cm.