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.
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
Please 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.
Useful 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 deals only with 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
Please 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.
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, curved. 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 a special case of a trapezoid. 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 lateral 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 / sinα
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 a rectangular one, in which one of the sides is perpendicular to the bases. 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 the total area of the figure.
Also, general formulas for calculating the area of a trapezoid are suitable for calculating the area of a rectangular trapezoid.
- If the lengths of the bases and the height (or 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.
The area of a curvilinear trapezoid is sought through the integral using the Newton-Leibniz formula:
This is how the areas of various types of trapezoids are calculated. 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.
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!
Best regards, 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 (equilateral) 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:
- The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.
- Isosceles trapezoids have equal sides and the angles they form with the bases.
- The midpoints of the diagonals of a trapezoid and the point of intersection of its diagonals are on the same straight line.
- If the sum of the sides of a trapezoid is equal to the sum of the bases, then a circle can be inscribed in it
- If the sum of the angles formed by the sides of a trapezoid at any of its bases is 90, then the length of the segment connecting the midpoints of the bases is equal to their half-difference.
- An isosceles trapezoid can be described by a circle. And vice versa. If a trapezoid fits into a circle, then it is isosceles.
- 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 understand and remember this formula as follows. As follows from the figure below, using the center line, a trapezoid can be converted into a rectangle, the length of which will be equal to half the sum of the bases.
You can also decompose any trapezoid into 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.