You will need

- To know the area of a trapezoid, the lengths of the bases and also the length of the middle line.

Instruction

1

In order to calculate the area of a trapezoid, you must use the following formula:

S = ((a+b)*h)/2, where a and b are the bases of the trapezoid, h is the height of the trapezoid.

In that case, if the area and lengths of bases are known, find the elevation by the formula:

h = (2*S)/(a+b)

S = ((a+b)*h)/2, where a and b are the bases of the trapezoid, h is the height of the trapezoid.

In that case, if the area and lengths of bases are known, find the elevation by the formula:

h = (2*S)/(a+b)

2

If the trapeze is known for its size and length of the middle line, then find its height will not be difficult:

S = m*h where m is the middle line here:

h = S/m.

S = m*h where m is the middle line here:

h = S/m.

3

In order to make both more understandable, you can give a couple of examples.

Example 1: length of the midline of the trapezoid is 10 cm, its area is 100 cm2. To find the height of the trapezoid is necessary to perform the action:

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 length of the bases equal to 8 cm and 12 cm find the height of the trapezoid need to run the action:

h = (2*100)/(8+12) = 200/20 = 10 cm

Answer: the height of this trapezoid is 20 cm

Example 1: length of the midline of the trapezoid is 10 cm, its area is 100 cm2. To find the height of the trapezoid is necessary to perform the action:

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 length of the bases equal to 8 cm and 12 cm find the height of the trapezoid need to run the action:

h = (2*100)/(8+12) = 200/20 = 10 cm

Answer: the height of this trapezoid is 20 cm

Note

There are several types of trapezoids:

An isosceles trapezoid is such a trapezoid in which the sides are equal.

Rectangular trapezoid is trapezoid, which has one internal angle equal to 90 degrees.

It should be noted that in a rectangular trapezoid height is equal to the length of a side at a right angle.

Around the trapezoid can be circumscribed by a circle, or enter her in this shape. To enter a circle only if the sum of its bases is equal to the sum of the opposite sides. To describe the circumference is possible only around an isosceles trapezoid.

An isosceles trapezoid is such a trapezoid in which the sides are equal.

Rectangular trapezoid is trapezoid, which has one internal angle equal to 90 degrees.

It should be noted that in a rectangular trapezoid height is equal to the length of a side at a right angle.

Around the trapezoid can be circumscribed by a circle, or enter her in this shape. To enter a circle only if the sum of its bases is equal to the sum of the opposite sides. To describe the circumference is possible only around an isosceles trapezoid.

Useful advice

A parallelogram is a special case of a trapezoid, because the definition of a trapezoid does not contradict the definition of a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel to each other. The trapezoid as in the definition it is only a couple of his parties. So every parallelogram is a trapezoid. The opposite is not true.

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

An isosceles trapezoid is a trapezoid whose nonparallel opposite sides are equal. Some formulas allow to find the area of a trapezoid with its sides, angles, height, etc. For the case of isosceles trapezoids the formula may be somewhat simplified.

You will need

- The formula for the area of a regular trapezoid

Instruction

1

The most common formula for calculating area of a trapezoid is S = (a+b)h/2. For the case of an isosceles trapezoid it explicitly does not change. We can only note that an isosceles trapezoid the angles at any of the bases are equal (DAB = CDA = x). As its sides is equal (AB = CD = C), then the height h can be calculate by the formula h = C*sin(x).

Then S = (a+b)*C*sin(x)/2.

Similarly, the area of a trapezoid can be recorded through the secondary side of the trapezoid: S = mh.

Then S = (a+b)*C*sin(x)/2.

Similarly, the area of a trapezoid can be recorded through the secondary side of the trapezoid: S = mh.

2

Consider a special case of an isosceles trapezoid, when its diagonals are perpendicular. In this case, on the property of a trapezoid, the height is equal to the sum of the bases.

Then the area of a trapezoid can be calculated by the formula: S = (a+b)^2/4.

Then the area of a trapezoid can be calculated by the formula: S = (a+b)^2/4.

3

Consider also another formula to find the area of the trapezoid: S = ((a+b)/2)*sqrt(c^2 - ((b-a)^2+c^2-d^2)/2(b-a))^2), where c and d are the sides of a trapezoid. Then, in the case of an isosceles trapezoid, when c = d, the formula becomes: S = ((a+b)/2)*sqrt(c^2-((b-a)^2/2(b-a))^2).

# Advice 3 : How to find the height of an isosceles trapezoid

The application of geometry in practice, especially in construction is obvious. A-line one of the most common geometric figures, the calculation accuracy of the elements that the beauty of the building.

You will need

- calculator

Instruction

1

Trapezoid is a quadrilateral with two sides parallel to the ground, and the other two not parallel sides. Trapezoid, the sides of which are equal, is called isosceles or ravnovesnoi. If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of bases, we consider the case when diagonals are not perpendicular.

2

Consider an isosceles trapezoid ABCD and describe its properties, but only those, knowledge of which will help us to solve the problem. From the definition of an isosceles trapezoid, the base AD = a is parallel to BC = b, and side AB = CD = c, it follows that the angles at the bases equal, that is, the angle BAQ = CDS = α, in the same way, the angle ABC = BCD = β. Summing up, it is fair to say that the triangle ABQ is equal to the triangle SCD, and hence, cut off AQ = SD = (AD – BC)/2 = (a – b)/2.

3

If in the problem statement we are given the lengths of the bases a and b and length of side C, the height h of the trapezoid is equal to the segment BQ is as follows. Consider the triangle ABQ, because, by definition, the height of the trapezoid is perpendicular to the base, it can be argued that the triangle ABQ rectangular. Side of the AQ triangle ABQ, based on the properties of an isosceles trapezoid, is given by AQ = (a – b)/2. Now knowing the two sides AQ and c by the Pythagorean theorem, find the height h. The Pythagorean theorem States that the hypotenuse squared is equal to the sum of the squares of the legs. We write this theorem applied to our problem: c^2=AQ^2+ h^2. It follows that h = √(c^2-AQ^2).

4

For example, consider the trapezoid ABCD in which the base AD = a = 10cm BC = b = 4 cm, side AB = c = 12cm. To find the height of a trapezoid h. Find the side of the AQ triangle ABQ. AQ = (a – b)/2 = (10-4)/2=3cm. Then substitute the values of the sides of the triangle in the Pythagorean theorem. h = √(c^2-AQ^2) = √(12^2-3^2) =√135=11.6 see

Useful advice

Properties of an isosceles trapezoid.

A straight line passing through the middle of the bases perpendicular to the bases is the axis of symmetry of the trapezoid.

Height, lowered from the top on the larger base, divides it into two segments, one of which is equal to the sum of the bases, the other half-difference basis.

In an isosceles trapezoid the angles at either base are equal.

In an isosceles trapezoid the length of the diagonals.

About an isosceles trapezoid can be circumscribed by a circle.

If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of the bases.

A straight line passing through the middle of the bases perpendicular to the bases is the axis of symmetry of the trapezoid.

Height, lowered from the top on the larger base, divides it into two segments, one of which is equal to the sum of the bases, the other half-difference basis.

In an isosceles trapezoid the angles at either base are equal.

In an isosceles trapezoid the length of the diagonals.

About an isosceles trapezoid can be circumscribed by a circle.

If in an isosceles trapezoid the diagonals are perpendicular, then the height is equal to the sum of the bases.

# Advice 4 : How to find side of a trapezoid if you know the base

Trapezium is a geometrical figure with four angles, two sides of which are parallel to each other and called bases, while the other two are not parallel are called lateral.

Instruction

1

Consider two tasks with different initial data.Task 1.Find the lateral

**side of the**isosceles**trapezoid**if you know*the base*BC = b,*the base*AD = d and the angle at the side of the BAD = alpha.Solution:Drop a perpendicular (the height**of the trapezoid**) from the vertex B to the intersection with the large*basis*m, will receive a cut BE. Write down the formula for AB using the measure of the angle: AB = AE/cos(BAD) = AE/cos(alpha).2

Find the AE. It would be the difference of the lengths of the two bases, divided in half. So: AE = (AD - BC)/2 = (d - b)/2.Now find AB = (d - b)/(2*cos(alpha)).In an isosceles

**trapezoid**the lengths of the sides are equal, hence CD = AB = (d - b)/(2*cos(alpha)).3

Task 2.Find the lateral

**side****of the trapezoid**AB, if you know the upper*base*BC = b; lower*base*AD = d; the height BE = h and the angle opposite the side of the CDA is equal to alpha.Solution:do a second height from vertex C to the intersection with the lower*basis*of m, will receive a cut of CF. Consider the right triangle CDF, find**the direction**FD according to the following formula: FD = CD*cos(CDA). The length of the side CD, find another formula: CD = CF/sin(CDA). So: FD = CF*cos(CDA)/sin(CDA). CF = BE = h, therefore, FD = h*cos(alpha)/sin(alpha) = h*ctg(alpha).4

Consider the right triangle ABE. Knowing the length of sides AE and be, you can find the third

**side**- the hypotenuse AB. You are aware of the length of the side BE find AE as follows: AE = AD - BC - FD = d - b - h*ctg(alpha).Using the following property of a right triangle - the square of the hypotenuse is equal to the sum of the squares of two - way AB:AB(2) = h(2) + (d - b - h*ctg(alpha))(2).The lateral sides**of the trapezoid**AB is the square root of the expression on the right side of the equality.# Advice 5 : How to find the height of a trapezoid if you know all sides

A trapezium is called a convex quadrilateral whose two opposite parallel sides non-parallel and the other two. If all opposite sides of a quadrilateral are pairwise parallel, then it is a parallelogram.

You will need

- - all side of the trapezoid (AB, BC, CD, DA).

Instruction

1

Non-parallel

**sides****of a trapezoid**are called lateral sides, and parallel bases. Line between the bases, perpendicular to them - the height**of the trapezoid**. If the lateral**sides****of a trapezoid**are equal, then it is called isosceles. First, consider the solution**of a trapezoid**that is not isosceles.2

Guide BE cut from point B to lower the base AD parallel to the side

**of the trapezoid**CD. Since BE and CD are parallel and held between the parallel bases**of the trapezoid is**BC and DA, then BCDE is a parallelogram, its opposite**sides**BE and CD are equal. BE=CD.3

Consider the triangle ABE. Calculate the direction of AE. AE=AD-ED. The base

**of the trapezoid**, BC and AD are known, and in the parallelogram BCDE opposite**sides**BC and ED are equal. ED=BC, then AE=AD-BC.4

Now find out the area of triangle ABE in the formula of Heron, calculating properiter. S=sqrt(p*(p-AB)*(p-BE)*(p-AE)). In this formula, p is properiter triangle ABE. p=1/2*(AB+BE+AE). To compute the area, you are aware of all necessary data: AB, BE=CD, AE=AD-BC.

5

Next, write down the area of the triangle ABE in another way - it is equal to half of the work of the triangle's height BH and

**the sides**AE, to which it is held. S=1/2*BH*AE.6

Express from this formula

*the height*of the triangle that is the height**of the trapezoid**. BH=2*S/AE. Calculate it.7

*If isosceles trapezoid, the solution is to do differently. Consider the triangle ABH. It is rectangular, as one of the corners, BHA, direct.*

8

Swipe from the top C

*height*CF.9

Examine the figure of the HBCF. HBCF rectangle, since two

**sides**are the height and the other two are the bases**of the trapezoid**, that is, the straight angles, and opposite**sides**are parallel. This means that BC=HF.10

Look at right triangles ABH and FCD. The angles at the elevation BHA and CFD are straight, and the angles at the lateral

**sides**x BAH and CDF are equal, since the trapezoid ABCD is isosceles, then the triangles are similar. Because of the height BH and CF are equal or the lateral**side**of the isosceles**trapezoid**, AB and CD are equal, then these triangles are equal. Hence, their**sides**AH and FD is also equal.11

Find AH. AH+FD=AD-HF. Because of the parallelogram HF=BC, and triangles, AH=FD, then AH=(AD-BC)*1/2.

12

Further, from the right triangle ABH Pythagoras to calculate

*the height*BH. The square of the hypotenuse AB is equal to the sum of the squares of the sides AH and BH. BH=sqrt(AB*AB AH*AH).# Advice 6 : How to calculate the height of a trapezoid

If in a quadrilateral only two opposite sides are parallel, it can be called a trapezoid. A pair of non-parallel lines to form this geometric shape, called sides, and another pair - bases. The distance between the two bases defines

**the height****of the trapezoid**and can be calculated in several ways.Instruction

1

If in terms of the lengths of both bases (a and b) and area (S) of the trapezoid, start calculating the height (h) find the half-sum of the lengths of the parallel sides: (a+b)/2. Then the resulting value divide the area - the result is the required size: h = S/((a+b)/2) = 2*S/(a+b).

2

Knowing the length of the middle line (m) and area (S), you can simplify the formula from the previous step. By definition the middle line of a trapezoid is equal to the sum of its bases, so to calculate the height (h) of the figure simply divide the area for the length of the midline: h = S/m.

3

You can determine the height (h) of the quadrilateral, and in that case, if given only the length of one of the sides (C) and the angle (α) formed by her and a long base. In this case, you should consider the triangle formed by this side, a height, and a short segment of the base, which cuts lowered to his height. The triangle is rectangular, known side will be the hypotenuse and the height is the leg. The ratio of the lengths of the leg and of the hypotenuse is equal to the sine of the opposite leg. of the angle, so to calculate the height of a trapezoid multiply the known side length to the sine of the known angle: h = s*sin(α).

4

The same triangle should be considered and if the length of the sides (C) and the angle (β) between it and the others (short) base. In this case, the angle between the side (the hypotenuse) and height (leg) will be 90° less is known of the conditions of an angle of β-90°. Since the ratio of the lengths of the leg and of the hypotenuse is equal to the cosine of the angle between them, the height of the trapezoid is calculate by multiplying the reduced cosine 90° angle to the length of the sides: h = C*cos(β-90°).

5

If a trapezoid inscribed in a circle of known radius (r), the calculation formula of height (h) is very simple and does not require knowledge of any other parameters. The circle, by definition, must touch each of the bases by only one point and these points will lie on the same line as the center of the circle. This means that the distance between them is equal to the diameter (twice the radius), held perpendicular to the bases, that is, coinciding with the height of a trapezoid: h=2*r.