The practice of last year's Unified State Exam 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 sharp and obtuse angles. Or in in some cases, at right angles. 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 form: 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 area formula 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 area formula 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 sharp cornerα. 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 = h2.
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 of a curvilinear trapezoid corresponds to the increment of the antiderivative on a given segment.
Examples of 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. in social networks. Let there be more good grades for the Unified State Examination and State Examinations!
blog.site, when copying material in full or in part, a link to the original source is required.
The section contains geometry problems (planimetry section) about trapezoids. If you haven't found a solution to a problem, write about it on the forum. The course will certainly be supplemented.
Trapezoid. Definition, formulas and properties
A trapezoid (from ancient Greek τραπέζιον - “table”; τράπεζα - “table, food”) is a quadrangle with exactly one pair of opposite sides parallel.A trapezoid is a quadrilateral whose pair of opposite sides are 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 lateral sides.
Trapezes are:
- versatile ;
- isosceles;
- rectangular
.Red and brown flowers The sides are indicated, and the bases of the trapezoid are indicated in green and blue.
A - isosceles (isosceles, isosceles) trapezoid
B - rectangular trapezoid
C - scalene trapezoid
A scalene trapezoid has all sides of different lengths and the bases are parallel.
The sides are equal and the bases are parallel.
The bases are parallel, one side is perpendicular to the bases, and the second side is inclined to the bases.
Properties of a trapezoid
- Midline of trapezoid parallel to the bases and equal to their half-sum
- A 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 a trapezoid cut off proportional segments from the sides of the angle (see Thales' Theorem)
- Point of intersection of trapezoid diagonals, the intersection point of the extensions of its sides and the middle of the bases lie on the same straight line (see also properties of a quadrilateral)
- Triangles lying on bases trapezoids whose vertices are the intersection point of its 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 lying on the sides trapezoids whose vertices are the intersection point of its diagonals are equal in area (equal in area)
- Into the trapeze 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 middle line in this case is equal to the sum of the sides divided by 2 (since the middle line of a trapezoid is equal to half the sum of the bases)
- Line segment, parallel to the bases and passing through the point of intersection 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)
Trapezoid angles
Trapezoid angles there are sharp, straight and blunt.Only two angles are right.
A rectangular trapezoid has two right angles, and the other two are acute and obtuse. Other types of trapezoids have two acute angles and two obtuse angles.
Obtuse angles of a trapezoid belong to the smaller along the length of the base, and spicy - 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 additionally constructing a trapezoid up to a triangle) some problems about trapezoids can be solved and some theorems can be proven.
How to find the sides and diagonals of a trapezoid
Finding the sides and diagonals of a trapezoid is done using the formulas given below:
In these formulas, the notations used are as in the figure.
a - the smaller of the bases of the trapezoid
b - the larger 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 lateral sides (Formula 2)
The practice of last year's Unified State Exam 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 form: 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 an acute angle α. 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 = h2.
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 of a curvilinear trapezoid corresponds to the increment of the antiderivative on a given segment.
Examples of 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 on social networks about this article. Let there be more good grades for the Unified State Examination and State Examinations!
website, when copying material in full or in part, a link to the source is required.
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.
Download (view) article material in *pdf format
That's all. Good luck to you!
Sincerely, Alexander.
What is an isosceles trapezoid? This geometric figure, whose opposite non-parallel sides are equal. There are several different formulas for finding the area of a trapezoid with different conditions, which are given in the tasks. That is, the area can be found if the height, sides, angles, diagonals, etc. are given. It is also impossible not to mention that for isosceles trapezoids there are some “exceptions”, thanks to which the search for area and the formula itself are significantly simplified. Below are detailed solutions for each case with examples.
Necessary properties for finding the area of an isosceles trapezoid
We have already found out that a geometric figure that has opposite, not parallel, but equal sides is a trapezoid, and an isosceles one. There are special cases when a trapezoid is considered isosceles.
- These are the conditions for equality of angles. So, a mandatory point: the angles at the base (take the picture below) must be equal. In our case, angle BAD = angle CDA, and angle ABC = angle BCD
- Second important rule– in such a trapezoid the diagonals must be equal. Therefore, AC = BD.
- Third aspect: the opposite angles of the trapezoid must add up to 180 degrees. This means that angle ABC + angle CDA = 180 degrees. The same applies to angles BCD and BAD.
- Fourthly, if a trapezoid allows a circle to be described around it, then it is isosceles.
How to find the area of an isosceles trapezoid - formulas and their descriptions
- S = (a+b)h/2 is the most common formula for finding the area, where A – lower base, b is the top base, and h is the height.
- If the height is unknown, then you can search for it using a similar formula: h = c*sin(x), where c is either AB or CD. sin(x) is the sine of the angle at any base, that is, angle DAB = angle CDA = x. Ultimately, the formula takes this form: S = (a+b)*c*sin(x)/2.
- The height can also be found using this formula:
- The final formula looks like this:
- The area of an isosceles trapezoid can be found through the midline and height. The formula is: S = mh.
Let us consider the condition when a circle is inscribed in a trapezoid.
In the case shown in the picture,
QN = D = H – the diameter of the circle and at the same time the height of the trapezoid;
LO, ON, OQ = R – radii of the circle;
DC = a – upper base;
AB = b – lower base;
DAB, ABC, BCD, CDA – alpha, beta – angles of the bases of the trapezoid.
A similar case allows the area to be found using the following formulas:
- Now let's try to find the area through the diagonals and the angles between them.
In the figure we denote AC, DB – diagonals – d. Angles COB, DOB – alpha; DOC, AOB – beta. Formula for the area of an isosceles trapezoid using the diagonals and the angle between them, ( S ) is:
