The many faces of a trapezoid. It can be arbitrary, isosceles or rectangular. And in each case you need to know how to find the area of a trapezoid. Of course, the easiest way to memorize basic formulas. But sometimes it's easier to use the one which is derived taking into account all the characteristics of specific geometric shapes.
A few words about the trapezoid and its members
Any quadrilateral whose two sides are parallel, can be called a trapezoid. In the General case they are not equal and are called bases. The larger of them — the lower and the other upper.
The other two sides are side. Arbitrary trapezoid they are of different lengths. If they are equal, the figure becomes isosceles.
If the angle between any side and the base will be equal to 90 degrees, then the trapezoid is rectangular.
All these features can help in solving the problem of how to find the area of a trapezoid.
Among the shape elements which can be invaluable in solving problems that can be identified such:
- the height, ie the segment perpendicular to both bases
- the middle line, which have at their ends the middle of the sides.
What is the formula to calculate the area, if you know the base and height?
This expression is given major because you can often learn these values, even when they are not given explicitly. So, to understand how to find the area of a trapezoid, you will need to fold both of the base and to divide them into two. The resulting value is then again multiplied by the height value.
If we denote the base letters and1 and2. the height h, the formula for the square will look like this:
The formula to calculate the area, if its height and the middle line
If you look closely to the previous formula, it is easy to see that it clearly has a value in the middle line. Namely, the sum of the bases divided by two. Let the middle line is marked with the letter l, then the formula for area will be:
The ability to find the square diagonally
This method will help, if you know the angle formed by them. Assume that the diagonal marked with the letter d1 d2. and the angles between them α and β. Then the formula to find the area of a trapezoid will be written in the following way:
In this expression can be easily replaced α by β. The result will not change.
How to find the area, if you know all of the figure?
There are situations when this figure is known side. This formula turns a cumbersome and difficult to remember. But it is possible. Let the sides have the designation: 1 and2. the base and1 more than the a2. Then the formula of the square will take the form:
Methods of calculating the area of an isosceles trapezoid
The first is that it is possible to inscribe a circle. And knowing its radius (it is designated by the letter r), and base angle — γ, you can use this formula:
S = (4 * r 2 ) / sin γ.
The latter General formula, which is based on knowledge of all sides of the figure, is substantially simplified due to the fact that the sides are of equal value:
Methods for calculating the area of a rectangular trapezoid
It is clear that you can use any of the following for arbitrary shapes. But sometimes it is useful to know about one of the features of such a trapezoid. It lies in the fact that the difference of the squares of the lengths of the diagonals is equal to the difference composed of the squares of the bases.
Often the formulas for a trapezoid are forgotten, while the expressions for the area of a rectangle and a triangle came to mind. Then we can apply a simple method. To divide a trapezoid into two pieces, if it is rectangular, or three. One will definitely be a rectangle, and the second, or two other triangles. After calculating the areas of these figures will remain their only folded.
This is a simple way to find the area of a rectangular trapezoid.
What if you know the coordinates of the vertices of a trapezoid?
In this case, you will need to use an expression that allows to determine the distance between points. It can be applied three times: in order to see both bases and the same height. And then just apply the first formula, which is described a little above.
To illustrate this method it is possible to result such example. Given the vertices with coordinates A(5
Before how to find the area of a trapezoid, the coordinates need to calculate the lengths of the bases. You will need this formula:
The upper base marked AB, so its length is equal to √<(8-5) 2 + (7-7) 2 > = √9 = 3. Bottom — SD = √ <(10-1) 2 + (1-1) 2 > = √81 = 9.
Now we need to make the height from the vertex to the base. Let her start will be at point A. the cut End will be on the under in the point with coordinates (5<(5-5) 2 + (7-1) 2 > = √36 = 6.
It remains only to substitute the resulting values into the formula area of a trapezoid:
S = ((3 9) / 2) * 6 = 36.
The problem is solved without units, because it is not the scale of a grid. It can be as millimeter and meter.
Examples of tasks
No. 1. Condition. The known angle between the diagonals of any trapezoid is equal to 30 degrees. The smaller diagonal has a value of 3 DM, and the second more it 2 times. It is necessary to calculate the area of a trapezoid.
Solution. First you need to know the length of the second diagonal, because without this will not count the answer. It is easy to compute, 3 * 2 = 6 (DM).
Now you need to use the appropriate formula for area:
S = ((3 * 6) / 2) * sin 30º = 18/2 * ½ = 4.5 (DM 2 ). The problem is solved.
Answer: the area of the trapezoid is equal to 4.5 DM 2 .
No. 2. Condition. In trapeze AVSD grounds are segments AD and BC. Point E the midpoint of side CD. Her held perpendicular to the line AB, the end of this segment is marked with the letter N. It is known that the lengths of AB and YONG are respectively 5 and 4 refer to the Need to calculate the area of a trapezoid.
Solution. First you need to make a drawing. Because the value of the perpendicular is less than the side to which it is held, then a-line will be a little stretched upwards. So YEON will appear inside the shape.
To clearly see the progress of the solution, you will need to perform additional construction. Namely, to draw a line, which is parallel to the side AB. The point of intersection of this line with AD — R, and with the continuation of the armed forces. WHRA the Resulting figure is a parallelogram. And its area is desired. This is due to the fact that the triangles obtained while additional construction equal. This follows from the equality side and two adjoining angles, one vertical, the other crosswise lying.
Find the area of a parallelogram can formula that contains the product of its side and height, descended on her.
Thus, the area of a trapezoid is equal to 5 * 4 = 20 cm 2 .
Answer: S = 20 cm 2 .
No. 3. Condition. Elements of an isosceles trapezoid have the following meanings: the lower base is 14 cm, the upper one is 4 cm, the acute angle is 45 °. We need to calculate its area.
Solution. Let the smaller base has a marking entirely. The altitude drawn from the point V, will be called EXT. Since the angle is 45°, then the triangle AVN will turn out rectangular and isosceles. So, an=VN. With an very easy to find. It is equal to half the difference of the bases. That is (14 – 4) / 2 = 10 / 2 = 5 (cm).
The base is known, the height is regarded. You can use the first formula, which here was considered for an arbitrary trapezoid.
S = ((14 4) / 2) * 5 = 18/2 * 5 = 9 * 5 = 45 (cm 2 ).
Answer: The required area is 45 cm 2.
No. 4. Condition. There is an arbitrary line of AWSD. At its sides from the same point O and E, so that OE parallel to the base AD. The area of trapezoid AED five times more than oats. To compute the value of OE, if you know the lengths of the bases.
Solution. You will need to carry out two parallel straight AB: the first through the point C, its intersection with ND — dot, MT
Unknown let OE=x. the height of the smaller trapezoid oats — n1. more AED — n2 .
Since the area of the two trapezoids are related as 1 to 5, we can write this equality:
The height and the sides of the triangles are proportional by construction. Therefore, we can write another equation:
In the last two entries in the left part are equal values, then we can write that (x and1 ) / (5(x and2 )) equal to (x – a2 ) / (a1 – x).
Here it is required to carry out a number of transformations. First multiply cross crosswise. Parentheses appear, which indicate the difference of the squares, after applying this formula we get a short equation.
It is necessary to reveal the brackets and move all terms with the unknown "x" to the left, and then take the square root.