# Area of a trapezium - formula and explanation

In today's article, we are going to show how to calculate the area of a trapezium. We have prepared several formulas for you so that you will be able to find the solution to the geometry problems concerning the trapezium area with different conditions specified in it. So, let's get started and make everything clear.

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## How to find the area of a trapezium

A trapezium is a convex quadrangle in which one pair of opposite sides is parallel to each other, and the other is not. Based on the definition of a trapezoid and features of a parallelogram, the parallel sides of the trapezoid cannot be equal to each other. Parallel opposite sides of a trapezoid are called its bases.

That is, the trapezoid has two bases. Non-parallel opposite sides of a trapezoid are called its sides. Depending on which sides, which angles they form with the bases, there are different types of trapezoids. Most often, trapezoids are divided into non-equilateral, isosceles and rectangular. So, how to find area of the trapezium?

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## Area of a trapezium rule

Formulas of the area of a trapezium include almost all of its elements, and the best solution is selected depending on the specified measures.

The main elements are height and median of a trapezium. The median is the line connecting the midpoints of the sides. The height of the trapezoid is drawn from the upper corner to the base. The area of the trapezoid through height is equal to the product of the half-sum of the base lengths multiplied by the height, and the formula for it is:

S = 1/2(a+b) • h. Here, S stands for the area, a and b stand for bases length of trapezium, and h stands for the height.

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If the median length is specified under the conditions of a problem, then this formula is greatly simplified, since it is equal to the half-sum of the base lengths: m = 1/2(a + b). Then the area of trapezium rule is:

S = m • h. Here, S stands for area, m stands for median length, and h stands for height.

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If lengths of all sides are given under the conditions of a problem, then we can consider an example of calculating the area of a trapezoid through this formula:

S = 1/2(a+b) • √cª - ((b-a)ª + cª - dª)/ 2(b-a))ª. Here. S stands for area, a and b stand for bases length, c and d stand for sides length, and ª = 2. Note, please, that the whole line "cª - ((b-a)ª + cª - dª)/ 2(b-a))ª" is standing under the square root.

For the isosceles trapezium, the area formula with specified bases is the following:

S = 1/2(a+b) • √cª - 1/4(b-a)ª. Here, S stands for area, a and b stand for the length of bases, c stands for the side length, and ª = 2. Note, please, that the whole line "cª - 1/4(b-a)ª" is standing under the square root.

If the length of diagonals of trapezium is given under the conditions of the problem, you can calculate the area of the trapezium using formula with diagonals and the smaller angle between them:

S = 1/2 • d¹ • d² • sinα. Here, S stands for area, and for diagonals length of the trapezium.

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A special case is the calculating of the area of an isosceles trapezium. The formula is derived in various ways - through the diagonal, through the angles adjacent to the base and the radius of the inscribed circle. If the length of the diagonals and the angle between them are specified under the conditions of the problem, you can use the following formula:

S = 1/2dª • sinα. Here, S stands for area, d stands for the length of diagonal, and ª = 2.

Next, let's consider another example of calculating the area of an isosceles trapezium. The formula using the sides and angles adjacent to the base makes it easy to find the area of the figure. That is, if bases, the side, and the angle are known under the conditions of the problem, one can easily calculate the area according to the following formula:

S = c • sinα(a - c • cosα) or S = c • sinα(b + c • sinα). Here, S stands for the area, c stands for the length of the side of isosceles trapezium, a or b stands for the length of the bases of trapezium.

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Another similar formula is using specified bases and angle adjacent to the bigger base of the trapezium. The formula is:

S = 1/2(bª - aª) • tgα. Here, S stands for the area, a and b for the length of bases, and ª = 2.

Also, the area of the trapezium can be calculated if the bases and angles adjacent to them are specified under the conditions of the problem. We can make calculations according to the following formula:

S = 1/2(bª-aª) • ((sina•sinβ)/sin(a+β)). Here, S stands for area, a and b stand for length of the bases, and ª = 2.

Area of isosceles trapezium can be also calculated if the length of the median and side and angle adjacent to the base are known under the conditions of the problem. In this case, the formula is the following:

S = m • c • sina. Here, S stands for area, m stands for the length of the trapezium, and c stands for the length of the side.

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Another formula for calculating the area of the isosceles trapezium can be performed with the help of the circle inscribed in it. If the radius of the circle and the angle adjacent to the base are known under the conditions of the problem, then we can apply the formula:

S = 4rª/sina. Here, S stands for area, r stands for the radius of the circles inscribed into the isosceles trapezium, and ª = 2.

This formula can be used with the diameter of the circle. Also, in this case, the diameter is equal to the height of the isosceles trapezium. The formula is the following:

S = Dª/sina. Here, S stands for the area, D stands for the diameter of the circle inscribed into the trapezium, and ª = 2. This formula is relevant for all types of trapezium providing it is possible to inscribe a circle into it.

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Here is another formula to find the area of the isosceles trapezium using the radius of the inscribed circle. If the radius of the circle is not known, we can calculate it using formula r = √ab/2, where a and b stand for the length of the bases of the trapezium (providing the length of the bases is specified under the conditions of the problem). And then, we can apply the formula to calculate the area:

S = r(a+b). Here, S stands for the area, a and b stand for the length of bases, and r stands for the radius of inscribed circle. This formula is relevant for all types of trapezium providing it is possible to inscribe a circle into it.

And, let's call it a day. We hope that our article has provided you with all the necessary information and answered your question on how to calculate area of a trapezium in many different ways. We are sure that now you can solve the mathematics problems like a pro and even give a lesson to someone.

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Source: Legit.ng

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