How to find the area of ​​a trapezoid?

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Before you find the area of ​​a trapezoid, you must give it definition.

Trapeze - a geometric shape with four corners at which two sides are parallel to each other, and the other two - no.The two sides that are parallel to each other, called bases, and non-parallel - side.If the parties, which are lateral, equal, an isosceles trapezoid will be called.If the intersection they form a right angle, it is rectangular.

In algebra, there is the concept of a curvilinear trapezoid - It is understood as a figure, bounded on one side of the x-axis, and the other - the graph of y = f (x) b and is defined on the interval [a;b]

How to find the area of ​​a trapezoid

calculated such geometrical figure by the formula S = 0,5 * (a + b) * h, where A and B are the length of the base of the trapezoid, and h - its height.

example.Dana trapezoid, the base of which one is 2 cm, the second - 3 cm, height - 4 cm. We expect the area under the formula, we get the result: S = 0, * 5 (2 + 3) * 4 = 12 cm2.

From the same formula that, knowing the area of ​​the figure, its height, the length of one side, you can find the length of the other.The second option - knowing the lengths of the sides and the area of ​​a trapezoid, you can find her height.

example.Dana Keystone, which has one base longer than the other 3 times.The height of figures - 3 cm, the area - 24sm2.Required to find the length of both bases.

decision.The area is calculated using the following formula S = 0.5 * (a + b) * h.Because of the problem is clear that one side more than the other 3 times, hence a = 3b.Replace in the formula and obtain S = 0.5 * (a + 3b) * h = 0.5 * 4B * h.As a result, we obtain S = 2c * h, i.e. a = S / 2h.Substitute numerical values ​​and get in = 6 cm, a = 18 cm.

However, this is not the only way that you can determine the area of ​​this figure.In the second method, before you find the area of ​​a trapezoid, it can be divided into simple geometric shapes: rectangle and two triangles (or a triangle, in the case of a rectangular trapezoid).In this case, the total area is calculated as the sum of the areas of these figures.As an option - you can insert it into a rectangle side of which will be equal to the length of the larger bases.In this case, the area of ​​the trapezoid is determined as the difference squares rectangles and triangles.

How to find the area of ​​a rectangular trapezoid?We have already mentioned that the rectangular trapezoid can be called a trapezoid, whose base (call it a) and side to intersect, forming a prima corner.Accordingly, in the above figure with avsd side will be high.Then, knowing the length of all three sides, you can find the area of ​​shapes S = 0,5 * (a + b) * c.

The simplest formula is as follows: S = a * h, where k - is the length of the midline of the trapezoid, h - its height.The problem is that in practice it is easier to measure the length of the base than to find the midline.And it is as follows:

Given: scalene, trapezoid AVSD non-rectangular, in which the sides AB and CD are the bases.Before you find the area of ​​a trapezoid, the next leg of the AU and the VD divided into 2 equal parts, marking the point of intersection of the letters T and C. Then the line GK, held parallel to the ground, and will be the middle line of the trapezoid m.

Another special case - where equilateral trapezoid.For it fit all of the formula (of course, except for a rectangular formulas).Its area can be determined by knowing the angle between the bases.The formula is as follows: S = (a + b) * c * sin (x) * 0.5, where A and B - the length of base side length c and x - the angle between them.

sometimes necessary to determine the area of ​​this figure has not only the geometry, but in algebra on the coordinate system.In this regard, students have the question of how to find the area of ​​a trapezoid in the coordinates.The principle of computation is the same - determines the length of the sides, as the difference of coordinates of points base height is calculated and the first formula is calculated area.The height will be considered a line drawn from the corner of one of the grounds to another base.

to determine the area of ​​a curvilinear trapezoid are integral.