Find the area of ​​the trapezoid if all sides are known. How to find the height of a trapezoid: formulas for all occasions

(S) trapezoid, start calculating the height (h) by finding half the sum of the lengths of the parallel sides: (a+b)/2. Then divide the area by the resulting value - the result will be the desired value: h = S/((a+b)/2) = 2*S/(a+b).

Knowing the length of the center line (m) and area (S), you can simplify the formula from the previous step. By definition, the midline of a trapezoid is equal to half the sum of its bases, so to calculate the height (h) of the figure, simply divide the area by the length of the midline: h = S/m.

It is possible to determine the height (h) of such a thing if only the length of one of the sides (c) and the angle (α) formed by it and the long base are given. In this case, one should consider the shape formed by this side, the height and the short segment of the base, which is cut off by the height lowered onto it. This triangle will be right-angled known side will be the hypotenuse in it, and the height will be the leg. The ratio of the lengths and the hypotenuse is equal to the angle opposite the leg, so to calculate the height of the trapezoid, multiply the known length of the side by the sine of the known angle: h = с*sin(α).

The same triangle is worth considering if the length of the side (c) and the magnitude of the angle (β) between it and the other (short) base are given. In this case, the angle between the side (hypotenuse) and the height (leg) will be 90° less than the angle known from the conditions: β-90°. Since the ratio of the lengths of the leg and hypotenuse is equal to the cosine of the angle between them, calculate the height of the trapezoid by multiplying the cosine of the angle reduced by 90° by the length of the side: h = с*cos(β-90°).

If a circle of known radius (r) is inscribed, calculating the height (h) will be very simple and will not require any other parameters. Such a circle, by definition, must have only one point at each of its bases, and these points will lie on the same line with the center. This means that the distance between them will be equal to the diameter (twice the radius) drawn perpendicular to the bases, that is, coinciding with the height of the trapezoid: h=2*r.

A trapezoid is a quadrilateral in which two sides are parallel and the other two are not. The height of a trapezoid is a segment drawn perpendicularly between two parallel lines. Depending on the source data, it can be calculated in different ways.

You will need

• Knowledge of the sides, bases, midline of a trapezoid, and also, optionally, its area and/or perimeter.

Instructions

Let's say there is a trapezoid with the same data as in Figure 1. Let's draw 2 heights, we get , which has 2 smaller sides by the legs of right-angled triangles. Let us denote the smaller roll as x. He is located

Geometry is one of the sciences that people encounter in practice almost every day. Among the diversity geometric shapes The trapezoid also deserves special attention. It is a convex figure with four sides, two of which are parallel to each other. The latter are called bases, and the remaining two are called sides. The segment perpendicular to the bases and determining the size of the gap between them will be the height of the trapezoid. How can you calculate its length?

Find the height of an arbitrary trapezoid

Based on the initial data, determining the height of a figure is possible in several ways.

Known area

If the length of the parallel sides is known, and the area of ​​the figure is also indicated, then to determine the desired perpendicular, you can use the following relationship:

S=h*(a+b)/2,
h – the desired value (height),
S – area of ​​the figure,
a and b are sides parallel to each other.
From the above formula it follows that h=2S/(a+b).

The value of the midline is known

If among the initial data, in addition to the area of ​​the trapezoid (S), the length of its midline (l) is also known, then another formula is useful for calculations. First, it’s worth clarifying what the midline is for this type of quadrilateral. The term defines the part of the straight line connecting the midpoints of the lateral sides of the figure.

Based on the trapezoid property l=(a+b)/2,
l – midline,
a, b – the base sides of the quadrilateral.
Therefore h=2S/(a+b)=S/l.

4 sides of the figure are known

In this case, the Pythagorean theorem will help. Having lowered the perpendiculars to the larger base side, use it for the two resulting right triangles. The final expression will look like:

h=√c 2 -(((a-b) 2 +c 2 -d 2)/2(a-b)) 2,

c and d – 2 other sides.

Angles at the base

If you have data on the base angles, use trigonometric functions.

h = c* sinα = d*sinβ,

α and β are the angles at the base of the quadrilateral,
c and d are its sides.

Diagonals of a figure and the angles that intersecting they form

The length of the diagonal is the length of the segment connecting the opposite vertices of the figure. Let us denote these quantities by the symbols d1 and d2, and the angles between them by γ and φ. Then:

h = (d1*d2)/(a+b) sin γ = (d1*d2)/(a+b) sinφ,

h = (d1*d2)/2l sin γ = (d1*d2)/2l sinφ,

a and b are the base sides of the figure,
d1 and d2 are the diagonals of the trapezoid,
γ and φ are the angles between the diagonals.

The height of the figure and the radius of the circle that is inscribed in it

As follows from the definition of this kind of circle, it touches each base at 1 point, which are part of one straight line. Therefore, the distance between them is the diameter – the desired height of the figure. And since the diameter is twice the radius, then:

h = 2 * r,
r is the radius of the circle that is inscribed in this trapezoid.

Find the height of an isosceles trapezoid

• As follows from the formulation, a distinctive characteristic of an isosceles trapezoid is the equality of its lateral sides. Therefore, to find the height of a figure, use the formula for determining this value in the case when the sides of the trapezoid are known.

So, if c = d, then h=√c 2 -(((a-b) 2 +c 2 -d 2)/2(a-b)) 2 = √c 2 -(a-b) 2 /4,
a, b – base sides of the quadrilateral,
c = d – its sides.

• If there are angles formed by two sides (base and side), the height of the trapezoid is determined by the following relationship:

h = c* sinα,
h = с * tgα *cosα = с * tgα * (b – a)/2c = tgα * (b-a)/2,

α – angle at the base of the figure,
a, b (a< b) – основания фигуры,
c = d – its sides.

• If the values ​​of the diagonals of the figure are given, then the expression for finding the height of the figure will change, because d1 = d2:

h = d1 2 /(a+b)*sinγ = d1 2 /(a+b)*sinφ,

h = d1 2 /2*l*sinγ = d1 2 /2*l*sinφ.

The practice of last year's Unified State Examination and State Examination shows that geometry problems cause difficulties for many schoolchildren. You can easily cope with them if you memorize all the necessary formulas and practice solving problems.

In this article you will see formulas for finding the area of ​​a trapezoid, as well as examples of problems with solutions. You may come across the same ones in KIMs during certification exams or at Olympiads. Therefore, treat them carefully.

What you need to know about the trapezoid?

To begin with, let us remember that trapezoid is called a quadrilateral in which two opposite sides, also called bases, are parallel, and the other two are not.

In a trapezoid, the height (perpendicular to the base) can also be lowered. The middle line is drawn - this is a straight line that is parallel to the bases and equal to half of their sum. As well as diagonals that can intersect, forming acute and obtuse angles. Or, in some cases, at a right angle. In addition, if the trapezoid is isosceles, a circle can be inscribed in it. And describe a circle around it.

Trapezoid area formulas

First, let's look at the standard formulas for finding the area of ​​a trapezoid. We will consider ways to calculate the area of ​​isosceles and curvilinear trapezoids below.

So, imagine that you have a trapezoid with bases a and b, in which height h is lowered to the larger base. Calculating the area of ​​a figure in this case is as easy as shelling pears. You just need to divide the sum of the lengths of the bases by two and multiply the result by the height: S = 1/2(a + b)*h.

Let's take another case: suppose in a trapezoid, in addition to the height, there is a middle line m. We know the formula for finding the length of the middle line: m = 1/2(a + b). Therefore, we can rightfully simplify the formula for the area of ​​a trapezoid to the following type: S = m* h. In other words, to find the area of ​​a trapezoid, you need to multiply the center line by the height.

Let's consider another option: the trapezoid contains diagonals d 1 and d 2, which do not intersect at right angles α. To calculate the area of ​​such a trapezoid, you need to divide the product of the diagonals by two and multiply the result by the sin of the angle between them: S= 1/2d 1 d 2 *sinα.

Now consider the formula for finding the area of ​​a trapezoid if nothing is known about it except the lengths of all its sides: a, b, c and d. This is a cumbersome and complex formula, but it will be useful for you to remember it just in case: S = 1/2(a + b) * √c 2 – ((1/2(b – a)) * ((b – a) 2 + c 2 – d 2)) 2.

By the way, the above examples are also true for the case when you need the formula for the area of ​​a rectangular trapezoid. This is a trapezoid, the side of which adjoins the bases at a right angle.

Isosceles trapezoid

A trapezoid whose sides are equal is called isosceles. We will consider several options for the formula for the area of ​​an isosceles trapezoid.

First option: for the case when a circle with radius r is inscribed inside an isosceles trapezoid, and the side and larger base form acute angleα. A circle can be inscribed in a trapezoid provided that the sum of the lengths of its bases is equal to the sum of the lengths of the sides.

The area of ​​an isosceles trapezoid is calculated as follows: multiply the square of the radius of the inscribed circle by four and divide it all by sinα: S = 4r 2 /sinα. Another area formula is a special case for the option when the angle between the large base and the side is 30 0: S = 8r2.

Second option: this time we take an isosceles trapezoid, in which in addition the diagonals d 1 and d 2 are drawn, as well as the height h. If the diagonals of a trapezoid are mutually perpendicular, the height is half the sum of the bases: h = 1/2(a + b). Knowing this, it is easy to transform the formula for the area of ​​a trapezoid that is already familiar to you into this form: S = h 2.

Formula for the area of ​​a curved trapezoid

Let's start by figuring out what a curved trapezoid is. Imagine a coordinate axis and a graph of a continuous and non-negative function f that does not change sign within a given segment on the x-axis. A curvilinear trapezoid is formed by the graph of the function y = f(x) - at the top, the x axis is at the bottom (segment), and on the sides - straight lines drawn between points a and b and the graph of the function.

It is impossible to calculate the area of ​​such a non-standard figure using the above methods. Here you need to apply mathematical analysis and use the integral. Namely: the Newton-Leibniz formula - S = ∫ b a f(x)dx = F(x)│ b a = F(b) – F(a). In this formula, F is the antiderivative of our function on the selected segment. And the area curved trapezoid corresponds to the increment of the antiderivative on a given segment.

Sample problems

To make all these formulas easier to understand in your head, here are some examples of problems for finding the area of ​​a trapezoid. It will be best if you first try to solve the problems yourself, and only then compare the answer you receive with the ready-made solution.

Task #1: Given a trapezoid. Its larger base is 11 cm, the smaller one is 4 cm. The trapezoid has diagonals, one 12 cm long, the second 9 cm.

Solution: Construct a trapezoid AMRS. Draw a straight line РХ through vertex P so that it is parallel to the diagonal MC and intersects the straight line AC at point X. You will get a triangle APХ.

We will consider two figures obtained as a result of these manipulations: triangle APX and parallelogram CMRX.

Thanks to the parallelogram, we learn that PX = MC = 12 cm and CX = MR = 4 cm. From where we can calculate the side AX of the triangle ARX: AX = AC + CX = 11 + 4 = 15 cm.

We can also prove that the triangle APX is right-angled (to do this, apply the Pythagorean theorem - AX 2 = AP 2 + PX 2). And calculate its area: S APX = 1/2(AP * PX) = 1/2(9 * 12) = 54 cm 2.

Next you will need to prove that triangles AMP and PCX are equal in area. The basis will be the equality of the parties MR and CX (already proven above). And also the heights that you lower on these sides - they are equal to the height of the AMRS trapezoid.

All this will allow you to say that S AMPC = S APX = 54 cm 2.

Task #2: The trapezoid KRMS is given. On its lateral sides there are points O and E, while OE and KS are parallel. It is also known that the areas of trapezoids ORME and OKSE are in the ratio 1:5. RM = a and KS = b. You need to find OE.

Solution: Draw a line parallel to RK through point M, and designate the point of its intersection with OE as T. A is the point of intersection of a line drawn through point E parallel to RK with the base KS.

Let's introduce one more notation - OE = x. And also the height h 1 for the triangle TME and the height h 2 for the triangle AEC (you can independently prove the similarity of these triangles).

We will assume that b > a. The areas of the trapezoids ORME and OKSE are in the ratio 1:5, which gives us the right to create the following equation: (x + a) * h 1 = 1/5(b + x) * h 2. Let's transform and get: h 1 / h 2 = 1/5 * ((b + x)/(x + a)).

Since the triangles TME and AEC are similar, we have h 1 / h 2 = (x – a)/(b – x). Let's combine both entries and get: (x – a)/(b – x) = 1/5 * ((b + x)/(x + a)) ↔ 5(x – a)(x + a) = (b + x)(b – x) ↔ 5(x 2 – a 2) = (b 2 – x 2) ↔ 6x 2 = b 2 + 5a 2 ↔ x = √(5a 2 + b 2)/6.

Thus, OE = x = √(5a 2 + b 2)/6.

Conclusion

Geometry is not the easiest of sciences, but you can certainly cope with the exam questions. It is enough to show a little perseverance in preparation. And, of course, remember all the necessary formulas.

We tried to collect all the formulas for calculating the area of ​​a trapezoid in one place so that you can use them when you prepare for exams and revise the material.

Be sure to tell your classmates and friends about this article. social networks. Let there be more good grades for the Unified State Examination and State Examinations!

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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 in terms of “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 came across 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 doubled 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 middle of CD, then (prove it yourself). 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.