Area Of Trapezium

Introduction

Area of trapezium is the region covered by a trapezium in a two-dimensional plane. A trapezoid is a convex quadrilateral with exactly one pair of parallel sides. A trapezoid is a two-dimensional shape. It looks like a table if we draw it on paper. A quadrilateral can be described as a polygon that has exactly four sides and four vertices in the terms of Euclidian geometry. The number of sides, corners of a trapezium is always four and it is always constant.

There are many examples of trapezoids in the real world. The fundamental use of trapezoids is the trapezoid rule, which divides the place beneath the curve into more than one trapezoids and evaluates the region of each trapezoid.

Trapezium

A parallelogram is additionally referred to as a trapezoid with two parallel sides.

From the figure above, we can see that sides AB and CD are parallel to each other, while AC and BD are non-parallel sides. Also, "h" is the distance between two parallel lines and represents the top of the trapezoid.

A trapezoid is a closed shape or polygon whose number of sides, vertices, and corners are exactly four. The pair of sides of a trapezium which are parallel is also its opposite sides.

There are many actual examples of trapezoids around us.

For example, a table with a trapezoidal surface.

Trapezoid types

Trapezoids are divided into three types. These three types are given below.

• Isosceles Trapezoid

• Scalene Trapezoid

• Right Angle Trapezoid

Scalene Trapezoid

Trapezoids with all sides that are unequal in length are called scalene trapezoids.

Right Angle Trapezoid

Right Angle Trapezoid is a trapezoid with at least two adjacent proper angles. A right angled trapezoid with at least one right angle.

Irregular Trapezoid

Normally, a trapezoid can be recognized when we see a quadrilateral with a pair of parallel sides are another pair of unparalleled but equal sides. These types of trapezoids are called the regular trapezoid. For the irregular trapezoids, the nonparallel sides are not equal. That is why they are called irregular trapezoids.

Trapezoid Properties

Some vital properties of the trapezoid are −

• In a trapezoid, there is a single pair of which are parallel and they are also opposite.

• Diagonals always meet each other. That is, they intersect.

• Non-equal lines of a trapezium are also non-parallel. Except in the case of isosceles, they are not equal.

• Middle section of the trapezoid is equal to the sum of parallel sides which is then halved.

Middle section of trapezoid

$\mathrm{Middle\: part =\frac{(AB+ CD )}{2}}$ where AB and CD are parallel sides or bases.

For an isosceles trapezoid, the legs or non-parallel sides are congruent. The sum of the interior angles of the trapezoid equals 360 degrees. ie

$$\mathrm{\angle A +\angle B +\angle C +\angle D = 360°}$$

The sum of two adjacent angles is 180°. This capacity that two adjacent angles are complementary.

Area of a Trapezium

The vicinity of a trapezoid can be discovered by means of taking the average of the two bases of the trapezoid and multiplying it with the aid of the height. Therefore, the region of the trapezoidal method is given by

$$\mathrm{Area\: of\: trapezoid,\: A = \frac{h(a+b)}{2}\: rectangular\: units.}$$

Where "a" and "b" are bases

"h" is peak or height.

Let the place of an isosceles trapezoid a and b be the lengths of the parallel sides of the trapezoid ABCD. This is the same as a being the length of the base and b being the size of the aspect parallel to a. And c is the length of the two non-parallel sides and h is the height of the isosceles trapezoid. where

$$\mathrm{AB = a,\: CD = b,\: BC = AD = c}$$

When h is subtracted vertically from CD and meets AB at E, a proper triangle AED is fashioned and vertical size

$$\mathrm{h = \sqrt{c^2 – (a-b)^2}[according\: to\: the\: Pythagorean\: theorem]}$$

As you know, the formulation for the location of a trapezoid is −

$$\mathrm{Area =\frac{1}{2} h(a+b)}$$

$$\mathrm{Area\: isosceles\: trapezoid = \frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$$

Solved Examples

1)What is the area of the trapezium if: the three sides are 3,4,5 respectively?

Ans. We know that the area will be given by the formula $\mathrm{\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$.

Hence, the area will be

$$\mathrm{=\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]= \frac{1}{2} [\sqrt{5^2 – (4-3)^2} (4+3)]=84}$$

So, the area of the trapezium will be 84.

2)What is the area of the trapezium if: the three sides are 5,6,7 respectively?

Ans. We know that the area will be given by the formula $\mathrm{\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$.

Hence, the area will be

$$\mathrm{=\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]= \frac{1}{2} [\sqrt{7^2 – (6-5)^2} (5+6)]=264}$$

So, the area of the trapezium will be 264.

3)What is the area of the trapezium if: the three sides are 7,8,9 respectively?

Ans. We know that the area will be given by the formula $\mathrm{\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$.

Hence, the area will be

$$\mathrm{=\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]= \frac{1}{2} [\sqrt{9^2 – (8-7)^2} (7+8)]=600}$$

So, the area of the trapezium will be 600.

4)What is the area of the trapezium if: the three sides are 10,11,12 respectively?

Ans. We know that the area will be given by the formula $\mathrm{\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$.

Hence, the area will be

$$\mathrm{=\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]= \frac{1}{2} [\sqrt{12^2 – (11-10)^2} (11+10)]=1501.5}$$

So, the area of the trapezium will be 1501.5.

5)What is the area of the trapezium if: the three sides are 7,3,2 respectively?

Ans. We know that the area will be given by the formula $\mathrm{\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]}$.

Hence, the area will be

$$\mathrm{=\frac{1}{2} [\sqrt{c^2 – (a-b)^2} (a+b)]= \frac{1}{2} [\sqrt{7^2 – (3-2)^2} (2+3)]=120}$$

So, the area of the trapezium will be 120.

Conclusion

Parallelogram is quadrilateral, which has two pairs of parallel sides. But, a trapezium is that quadrilateral that has only a single pair of parallel sides. These parallel sides are obviously placed opposite to each other. The largest parallel side is usually considered as the base of the trapezium.

FAQs

1.Does a trapezoid have parallel sides?

A trapezium is that quadrilateral which has only a single pair of parallel sides. These parallel sides are obviously placed opposite to each other.

2.Can we tell that the diagonals of the trapezoids are equal?

The diagonals of the trapezoids need to not be the same. A trapezoid has a single pair of parallel sides. However, in order that the quadrilateral has equal diagonals, the two units of lines must be parallel, such as for a square, rectangle etc.

3.In trapezoids, do the diagonals intersect at each other’s midpoints?

The diagonals of the trapezoid do not intersect at each other’s midpoints. If the diagonal bisects, the trapezoid will become a parallelogram. So all parallelograms are trapezoids, but not all trapezoids are parallelograms.

4.Is a square a trapezoid?

Yes, a rectangle has to have at least one pair of parallel sides to be a trapezoid, the other two lines may or may not be parallel that doesn’t matters.

5.Is the kite a trapezoid?

No, a kite is not a trapezoid, since it doesn’t have a pair of parallel sides.