You will need

- The length of the major base, the midline, the height of a trapezoid, area of trapezoid

Instruction

1

The easiest way to find the small base, knowing the large base

**of the trapezoid**and its mid. Property**of a trapezoid**, its middle line is equal to the sum of the bases. Then the small base**of the trapezoid**can be expressed as: b = 2m-a, where m is the middle line**of a trapezoid**, a large base**of the trapezoid**.2

If you know the area

**of a trapezoid**, the height and length of a large base, it is enough to find the small base. According to the formula area**of a trapezoid**S = h(a+b)/2. Therefore, b = (2S/h)-a.3

Let the trapezoid ABCD is angled (as shown). Then its small base can be calculated through a large, elevation and angles with a large base (let's call them x and y).

In this case, the length of the small base can be represented using this data is: b = a-h*(ctg(x)+ctg(y)).

In this case, the length of the small base can be represented using this data is: b = a-h*(ctg(x)+ctg(y)).

4

Now let the notch-line (assume that the angle y is stupid). In this case, the small base can be expressed as: b = a-h(ctg(x)-ctg(180-y)).

# Advice 2 : How to find the diagonal of an isosceles trapezoid

Trapezoid, where the length of sides equal and base parallel, is called isosceles or isosceles. Both diagonals in such a geometrical figure have the same length, which is depending on known parameters of the trapezoid can be calculated many different ways.

Instruction

1

If you know the lengths of the bases of an isosceles trapezoid (A and B) and the length of its sides (C), to determine the lengths of the diagonals (D) we can use the fact that the sum of the squares of the lengths of all sides equals the sum of the squares of the lengths of the diagonals. This property stems from the fact that each of the diagonals of a trapezoid is the hypotenuse of a triangle, the legs of which serve as sidewall and the base. And according to Pythagoras the sum of the squares of the lengths of the legs is equal to the squared length of the hypotenuse. As the sides in an isosceles trapezoid are equal and its diagonals, then this property can be written in the form: A2 + B2 + 2C2 = 2D2. From this formula it follows that the length of the diagonal is equal to the square root of half the sum of the squares of the lengths of the bases, folded with the square of the length of the sides: D = √((A2 + B2)/2 + C2).

2

If the lengths of the sides is not known, but is the length of the middle line (L) and height (H) of an isosceles trapezoid, the length of the diagonal (D) is also easy to calculate. Since the length of the middle line is equal to the sum of the bases of the trapezoid, it gives us the opportunity to find the length of a line between a point on a larger basis, which lowered the height, and the top adjacent to the ground. In an isosceles trapezoid the length of this segment will coincide with the length of the middle line. Since the diagonal closes this segment and the height of the trapezoid into a right triangle, calculate its length is not difficult. For example, according to the same Pythagorean theorem it is equal to the square root of the sum of the squares of the height and the middle line: D=√(L2 + H2).

3

If you know the lengths of both bases of an isosceles trapezoid (A and B) and its height (H), then, as in the previous case, we can calculate the length of a line between point lowered on the big side elevation and the adjacent top. The formula from the previous step transformirovalsya like the following: D=√((A + B)2/4 + H2).

# Advice 3 : How to find the side of a rectangular trapezoid

Each trapezoid has two sides and two bases. In order to find area, perimeter and other parameters of this shape, you need to know at least one of the sides. Often under the terms of the task required to find the side of a rectangular trapezoid.

Instruction

1

Draw a rectangular trapezoid ABCD. The sides of this figure indicate, respectively, as AB and DC. The first side DC coincides with the height of the trapezoid. It is perpendicular to two rectangular bases of the trapezoid.

There are several ways of finding the sides. So for example, if the task is given to the second lateral side BA and the angle ABH=60, then the height of the first find the most simple of ways, having a height of BH:

BH=AB*sinα

Since BH=CD, then CD=AB*sinα=√3AB/2

There are several ways of finding the sides. So for example, if the task is given to the second lateral side BA and the angle ABH=60, then the height of the first find the most simple of ways, having a height of BH:

BH=AB*sinα

Since BH=CD, then CD=AB*sinα=√3AB/2

2

If, on the contrary, given of the trapezoid is denoted as CD, and you want to find her the same side of AB, this problem is solved in some other way. Since BH=CD and at the same time, BH is a leg of the triangle ABH, we can conclude that the side AB is equal to:

AB=BH/sinα=2BH/√3

AB=BH/sinα=2BH/√3

3

The problem can be solved and in that case, if the angle is unknown, provided that the two bases and the side AB. However, in this case, you can only find the side CD, which is the height of the trapezoid. Initially, knowing the value of the grounds, find the length of line AH. It is equal to the difference between the larger and the smaller bases, because we know that BH=CD:

AH=AD-BC

Then, using the Pythagorean theorem, find the height BH is equal to the side CD:

BH=√AB^2-AH^2

AH=AD-BC

Then, using the Pythagorean theorem, find the height BH is equal to the side CD:

BH=√AB^2-AH^2

4

If a rectangular trapezoid has the diagonal BD and angle 2α, as shown in figure 2, the side AB can also be found by the Pythagorean theorem. To do this, first calculate the length of base AD:

AD=BD*cos2α

Then find the side AB in the following way:

AB=√BD^2-AD^2

Then prove the similarity of triangles ABD and BCD. Since these triangles share one common side - diagonal, and thus, two angles are equal, as can be seen from the figure, these shapes are similar. Based on this evidence, find the second lateral side. If you know the upper base and the diagonal, you find the normal way using a standard spherical law of cosines:

c^2=a^2+b^2-2ab cos α, where a, b, C be the sidelengths of a triangle, α is the angle between sides a and b.

AD=BD*cos2α

Then find the side AB in the following way:

AB=√BD^2-AD^2

Then prove the similarity of triangles ABD and BCD. Since these triangles share one common side - diagonal, and thus, two angles are equal, as can be seen from the figure, these shapes are similar. Based on this evidence, find the second lateral side. If you know the upper base and the diagonal, you find the normal way using a standard spherical law of cosines:

c^2=a^2+b^2-2ab cos α, where a, b, C be the sidelengths of a triangle, α is the angle between sides a and b.