Trapezium-01

Trapezium

Definition of Trapezium

A trapezium is a convex quadrilateral or four-sided polygon with one pair of parallel sides of unequal lengths and one pair of non-parallel sides.

The parallel sides of a trapezium are called bases and the non-parallel sides of a trapezium are called legs or the lateral sides.

The perpendicular distance between the parallel sides of a trapezium is known as the height or altitude of the trapezium.

WHAT ARE QUADRILATERALS?

A quadrilateral is a closed plane figure bounded by four-line segments. It has four sides, four vertices, four angles, and two diagonals. Quadrilaterals are either simple or complex.

A simple quadrilateral is either convex or concave.

  1. Convex quadrilateral: If each of the interior angles of a quadrilateral measures less than 180°, then the quadrilateral is a convex quadrilateral.
  2. Concave Quadrilateral: A quadrilateral is concave if at least one of the measures of the angles more than 180° and one diagonal of the quadrilateral lies outside it.

There are different types of quadrilaterals.

  • Parallelogram
  • Rectangle
  • Square
  • Rhombus
  • Trapezium
  • Kite

TYPES OF TRAPEZIUM:

Isosceles trapezium: An isosceles trapezium is defined as the trapezium in which nonparallel sides of the trapezium are equal and form equal angles with the bases.

trapezium 1

Scalene Trapezium: A scalene trapezium is defined as the trapezium in which all the sides and angles are of different measures.

trapezium 2

Right-angled Trapezium: A right-angled trapezium is defined as the trapezium which has one pair of right angles and in which one pair of opposite sides are parallel to each other.

trapezium 3

PROPERTIES OF TRAPEZIUM:

trapezium 4

  1. The trapezium has four sides AB, BC, CD and DA and four angles A, B, C and D
  2. It is a two-dimensional figure like a rectangle, parallelogram, square.
  3. One pair of Parallel sides AB and CD should be opposite to each other or the bases of the trapezium are parallel to each other i.e., AB parallel to CD.
  4. One pair of non-parallel sides i.e., AD and BC.
  5. The sum of interior angles is 360°
  6. i.e., A + B + C + D = 360°
  7. Sum of the angles of adjacent sides of the trapezium is supplementary or 180°
  8.        i.e., A + D = 180° and B + C = 180°
  9. Diagonals of a trapezium bisect each other.
  10. No sides, angles and diagonals are congruent
  11. Except for an isosceles trapezium, a trapezium has non-parallel sides unequal
  12. A line drawn from the mid of non-parallel sides is called the mid-segment.
  13. The line that joins the mid-point of the non-parallel sides is always parallel to the bases.
  14. The mid-points of sides are collinear to the intersection point of diagonals. 
  15. The trapezium will get divided into two unequal parts if one cuts it into two sides from the mid of non-parallel sides.

AREA OF A TRAPEZIUM

The area of trapezium is the region covered by the trapezium in two-dimensional geometry. The area of trapezium is equal to the product of half of the sum of parallel sides and distance between the parallel sides. The area of trapezium is measured in square units.

Area of a Trapezium =1/2 × (Sum of parallel sides) × Distance between the parallel sides

trapezium 5

A=1/2 × (a + b) × h

Where, a and b are length of parallel sides, h is the distance between the parallel sides.

PERIMETER OF TRAPEZIUM

The perimeter of a trapezium is defined as the total length of the boundary of the shape, i.e., the sum of all its sides.

Perimeter of a Trapezium = Sum of all the sides = a + b + c + d

where, a, b, c, and d, are the sides of the trapezium.

MEDIAN OF A TRAPEZIUM

The median (also called a midline or midsegment) is a line segment half-way between the two bases.

The median's length is the average of the two base lengths:

m = (a + b)/2

WHAT IS THE DIFFERENCE BETWEEN TRAPEZIUM AND TRAPEZOID?

In US, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid.

In UK, it is considered the opposite, a trapezoid is considered to be a quadrilateral with no parallel sides.

The Trapezium is a four-sided shape and a two-dimensional figure with exactly one pair of parallel sides opposite each other.

whereas, a trapezoid is also a four-sided shape with no pair of parallel sides opposite each other.

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Examples

Q1 . Find the area of the Trapezium, in which the sum of the parallel sides is 70cm, and its height is 10cm.

Solution:

Given, the sum of parallel sides 70cm and height, h=10cm

We know that,

Area of a trapezium, 

A=1/2×Sum of parallel sides × Distance between the parallel sides

Substitute the given values,

A=1/2×70×10

A=35×10

A=350cm2

Therefore, the area of Trapezium =350cm2

 

Q2. The area of a trapezium is 140 cm² and its height is 10 cm. If one of the parallel sides is 8cm, find the other parallel side?

Solution:

Given that,

Area of the trapezium = 140 cm²

Height = 10 cm

 

One parallel side of the trapezium, a is 8cm, let another parallel side is b.

We know that, Area of a trapezium, 

 

A=1/2×Sum of parallel sides × Distance between the parallel sides

 

Substitute the given values in formula

140 = ½ x (8 + b) x 10

140 = 5 x (8 + b)

28 = 8 + b

So, b = 28 – 8 = 20cm

Therefore, the other parallel side is 20cm.

Solution:

Q3.: Find the height of trapezium if parallel sides are 6cm and 8cm and area of trapezium is 21cm2.

Given, the parallel sides are 6cm and 8cm

So, a = 6cm and b = 8cm

And Area = 21cm2  

Area of trapezium A=1/2 × (a + b) × h

Where, a and b are length of parallel sides, h is the distance between the parallel sides.

Substitute the values in the formula,

21 = ½ x (6+ 8) x h

21 = ½ x 14 x h

21 = 7 x h

So, h = 21/7 = 3cm

Therefore, height of trapezium is 3cm.

Q4 . Find the perimeter and area of the trapezium whose parallel sides are 12 cm and 20 cm. The distance between the bases is 40 cm and the non-parallel side length is 15 cm each.

Solution:

Given, Length of parallel sides = 12 cm and 20 cm

The distance between the bases, i.e., height= 40 cm

Area=1/2 x (a+ b) ×h

= ½ x (12 + 20) × 40

= 20 x 32

= 640cm2

Perimeter = Sum of all the sides

= 12 + 20 + 15 + 15

= 62 cm

Hence, the area and perimeter are 640cm2 and 62 cm respectively.

Q5.The area of trapezium is 18sq.cm. Its height and base are 3cm and 5cm respectively. Find the length of the side parallel to the base.

Solution:

Given, the height is 3cm and base is 5cm

So, h = 3cm and b = 5cm

And Area = 18cm2  

Area of trapezium A=1/2 × (a + b) × h

Where, a and b are length of parallel sides, h is the distance between the parallel sides.

Substitute the values in the formula,

 

18 = ½ x (a + 5) x 3

36 = 3a + 15

3a = 36 – 15

3a = 21

So, a = 7cm

Therefore, length of the side parallel to the base is 7cm.

 

Q6. What is the median of the trapezium if the lengths of the bases are 4cm and 10cm?

Solution:

The median of a trapezium is the average of the two base lengths:

m = (a + b)/2

Here, the lengths given are 4cm and 10cm

So, m= (4 + 10)/2

        = 14/2

         = 7cm.

Practice Questions

Trapezium Practice Questions

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Attend Trapezium Quiz Questions

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The trapezium has how many sides

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Parallel sides of a trapezium is called as?

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sum of interior angles is

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Line drawn from the mid of non-parallel sides is called as

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What are the Types of Trapezium

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Perimeter of Trapezium is

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FAQ

What are the different types of a quadrilateral?

 The different types of quadrilaterals are Rectangle, Square, Parallelogram, Trapezium or Trapezoid, and Rhombus.

What is a trapezium?

A Trapezium is a quadrilateral which has one pair of opposite sides are parallel and the other pair of sides are non-parallel.

 

What is the area of a trapezium?

 Area of Trapezium is A=1/2 × Sum of parallel sides × Distance between the parallel sides

What is the perimeter of a trapezium?

Perimeter of a trapezium is sum of all the sides of the trapezium. For e.g. – AB +BC+CD+DA

What are the three types of trapezium?

The three types of trapezium are:

  • Isosceles Trapezium 
  • Scalene Trapezium
  • Right - angled Trapezium



  • Sin = Opposite/Hypotenuse
  • Cos = Adjacent/Hypotenuse
  • Tan = Opposite/Adjacent

Cot = 1/Tan = Adjacent/Opposite

Cosec = 1/Sin = Hypotenuse/Opposite

Sec = 1/Cos = Hypotenuse/Adjacent

What is a Scalene trapezium?

A scalene trapezium is defined as a trapezium in which all the sides and angles are different (not equal to each other)



What is an isosceles trapezium?

An isosceles trapezium is defined as a trapezium where the sides have the same length and the base angles are equal

Are the diagonals of a trapezium equal?

No, the diagonals of a trapezium are not equal because all the four sides are not equal in a trapezium.



Is trapezium a parallelogram?

A trapezium is not a parallelogram because a parallelogram has two pairs of parallel sides. But a trapezium only has one pair of parallel sides.

What is the trapezium in the U.S. called?

The Trapezium in the U.S. is called a Trapezoid.

In a trapezium ABCD, if AB || CD, then which pair of angles are supplementary?

A and D, B and C are supplementary pairs of angles. So, A + D = 180° and B + C = 180°

Do congruent trapeziums have unequal area?

No, Congruent trapeziums have equal area.



What is the sum of the angles of a trapezium?

The sum of the angles of a trapezium is 360°.

What is the median of the trapezium?

The median is a line segment half-way between the two bases. It is also called a midline or midsegment.

What are the properties of trapezium?

The properties of trapezium are:

  1. Bases are parallel to each other.
  2. Adjacent angles are supplementary to each other.
  3. Only one pair of opposite sides are parallel.
  4. A line drawn from the mid of non-parallel sides is called the mid segment.
  5. The sum of all the interior angles in a trapezium is always 360°.

What is the formula of median of a trapezium?

The median's length is the average of the two base lengths:

m = (a + b)/2

where a and b are the lengths of the two bases

   
Acute Angled TriangleRight Angled TriangleObtuse Angled Triangle

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