What is a Trapezium?Â
A trapezium is a quadrilateral with one set of parallel, opposite sides. The non-parallel sides of a trapezium are called its legs, and the parallel sides are referred to as bases. Another name for a trapezium is a trapezoid.Â
The term trapezium is used for a shape having two parallel sides which is sometimes referred to as parallelogram as well. But the difference lies between the length of those sides. In a parallelogram, 2 opposite sides are of equal length. Whereas in a trapezium, all sides can have different lengths.
In the above figure, sides AB and CD run parallel to each other, while sides AC and BD do not. The height of the trapezium is shown by the distance, or "h," between the two parallel sides.
Trapezium's ShapeÂ
Like other 2D forms, the trapezium is a two-dimensional structure. First, it is formed by joining four straight lines to create four distinct angles. A trapezium only has one pair of parallel lines that cross each other, whereas squares and rectangles have two pairs of parallel lines.
A trapezium, like other polygons, has four sides and four corners in a closed shape. The two lines in the figure are parallel to one another and opposed to one another. This shape can be thought of as a trapezium-shaped table with a surface.
Trapezium TypesÂ
A trapezium can be classified into three different types based on its structure:
- Isosceles Trapezium
- Scalene Trapezium
- Right Trapezium
Isosceles Trapezium An isosceles trapezium has all of its non-parallel sides or legs with the same length.
Scalene Trapezium A scalene trapezium has all sides and angles with different measurements.
Right Trapezium A right trapezium features two or more right angles next to each other.
Irregular Trapezium We know that a trapezium has two non-parallel sides in addition to exactly one set of parallel sides. In a regular trapezium, the two non-parallel sides will be equal, while the other two non-parallel opposing sides in an irregular trapezium will be unequal.
Properties of TrapeziumÂ
- Here are some key characteristics of a trapezium:
- A trapezium has only one set of opposite sides that are parallel.
- Except for the isosceles trapezium, the diagonals intersect each other.
- The trapezium’s non-parallel sides are uneven.
- The line that connects the midpoints of the non-parallel sides is always parallel to the bases or parallel sides and is equal to half the sum of the parallel sides.
- Mid-section = (AB + CD) / 2 where the parallel sides, or bases, are CD and AB.
- The legs or non-parallel sides of an isosceles trapezium are congruent.
- The sum of a trapezium’s inner angles is 360 degrees. That is, 360° =
- The sum of the two adjacent angles is 180°, meaning the two adjacent angles are supplementary.
Formulas for TrapeziumÂ
Here are the two key trapezium formulas:
- Trapezium Area: ½ (Sum of parallel sides) × (Distance between parallel sides)
- The perimeter of a Trapezium: The total of all four sides
Area of a TrapeziumÂ
We calculate a trapezium’s area by multiplying its height by the average of its two bases. Thus, the trapezium formula's area is: The area of a trapezium equalsÂ
A= h(a + b) / 2 square units. Where:
- "a" and "b" are the two bases
- "h" denotes the height or altitude.
Isosceles Trapezium AreaÂ
If "a" is the base length and "b" is the side parallel to "a," then "a" and "b" are the lengths of the parallel sides of a trapezium ABCD. Therefore, a > b
Here, "h" is the height of the isosceles trapezium, and "c" is the length of the two non-parallel sides. Thus:
If we draw a perpendicular "h" from CD to meet AB at E, we form a right triangle AED. The perpendicular length, h = √(c² – (a – b)²) based on Pythagoras' theorem
We use the area of a trapezium formula: Area equals ½ h(a + b). Thus, the isosceles trapezium's area equals ½ [√(c² – (a – b)²) × (a + b)].
Perimeter of a TrapeziumÂ
We calculate a trapezium's circumference by summing its sides. Thus, the perimeter formula is: The trapezium's perimeter, P = a + b + c + d units Where:
"a," "b," "c," and "d" denote the trapezium's sides.
Isosceles Trapezium's PerimeterÂ
In an isosceles trapezium, if "a," "b," and "c" are the lengths of the parallel sides and "c" is the length of the two non-parallel sides, the perimeter will be: A + B + C = Perimeter
For instance, if a normal trapezium has parallel sides measuring 10 and 12 cm in length, and non-parallel sides measuring 5 cm each, calculate the perimeter. With a = 10 cm, b = 12 cm, and c = 5 cm, we apply the perimeter formula: 10 + 12 + 2 × 5 = a + b + 2c Thus, P = 10 + 12 + 10 = 32 cm.
How to Determine Trapezium Angles? Examples
In a regular or isosceles trapezium, the angles where parallel lines meet are equal. We know that the sum of all internal angles of a quadrilateral equals 360 degrees.
To find the sum of the two angles on the side opposite an angle x between one parallel and one non-parallel side, subtract twice this angle from 360. Divide the result by two to find the measure of the fourth angle.
Example 1: if the base angle of a trapezium between a parallel and a non-parallel side is 72 degrees, determine the angle opposite it.
Answer: Assuming a trapezium ABCD has a base angle of 72 degrees, let x be the angle we find. The total of all the angles in the quadrilateral equals 360 degrees.
Subtract 2 × 72 from360= 360 – 144 = 216
Thus, the total of the two angles mentioned above is 216. Given that the two aforementioned angles in an isosceles trapezium are equal:Â
x = 216/2 = 108
Example 2: Find the fourth side of the trapezium, if the other three sides are 8 cm, 12 cm, and 16 cm, and the perimeter is 40 cm.:Â
Ans: Perimeter is given as the sum of all its sides. Let the length o unknown be ‘x’ units.Â
Perimeter =Â 40Â
40 = 8 + 12 + 16 + x
x = 40 – (8 + 12 + 16)Â
  = 4 cmÂ
Thus, the length of the unknown side is 4 cm
Example 3: A trapezium has parallel sides of lengths 15 cm and 11 cm, and non-parallel sides of length 5 cm each. Calculate the perimeter of the trapezium.
Solution:
It is a Isosceles Trapezium because it is clearly mentioned that non parallel sides of length 5 cm each are equal.
According to Isosceles Trapezium if two non-parallel sides of Trapezium are of equal length then it is known as Isosceles Trapezium.
Given,
a = 15 cm
b = 11 cm
c = 5 cm
Perimeter = a + b + 2cÂ
P = 15 + 11 + 2(5)
P = 15 + 11 + 10
P = 36 cm
Example 4: Find Perimeter of a Trapezium whose sides are 12 cm, 14 cm, 16 cm, and 18 cm.
Solution:
P = Sum of all Sides
P = 12 + 14 + 16 + 18
P = 60 cm
Hence, perimeter of trapezium is 60 cm
Example 5: Find Area of Trapezium, in which the sum of parallel sides is 60 cm, and its height is 10 cm.
Solution:
Given,
Sum of Parallel sides 60 cm
height, h = 10 cm
Area of Trapezium, A = 1/2 × Sum of parallel sides × Distance between Parallel Sides
Substituting Given Values,
A =1/2×60×10
A = 30×10
A = 300 cm2
Therefore, Area of Trapezium =300 cm2
FAQ
1. What characteristics does a trapezium have?Â
A trapezium is a two-dimensional shape. It has one set of parallel sides called "bases."
2. What distinguishes a trapezium from a trapezoid?Â
A trapezium is a four-sided polygon with two parallel sides and two non-parallel sides. We use "trapezium" to describe quadrilaterals with non-parallel sides.
3. What features distinguish a trapezium?
A trapezium is a quadrilateral with one set of parallel sides and non-parallel sides called legs. It also has two diagonals or lines connecting the midpoints of opposing sides.
4. What is the definition of a trapezium?Â
A trapezium is a quadrilateral with one set of parallel sides. We call the other two sides the "base," and the parallel sides the "legs."
5. How do a rhombus and a trapezium differ from one another?Â
A trapezium is a quadrilateral with at least one pair of opposite, non-parallel sides, and two sets of parallel sides. A rhombus is a quadrilateral with four congruent sides, two pairs of equal adjacent angles, and two pairs of equal opposite angles.
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