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Maths > Mensuration > Trapezium, Parallelogram and Rhombus
Mensuration

Trapezium, Parallelogram and Rhombus

A quadrilateral is a closed figure with four sides, four angles and four vertices. There are different types of quadrilaterals. They are Trapezium, Parallelogram and Rhombus. Let us learn about these different types of quadrilaterals.

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Trapezium

A trapezium is a quadrilateral wherein one pair of the opposite sides are parallel while the other isn’t. This 4 sided closed flat shape is also referred to as a trapezoid. In other words, a trapezium is basically a triangle with the top sliced off. The parallel sides of a trapezium are called the base.

Trapezium ABCD

Figure ABCD is a trapezium. Here AB||CD but yes AD and BC are not parallel.

To calculate the height we need to draw a perpendicular from one parallel side to another.

The area of a trapezium can be calculated by taking the average of the two bases and multiplying it by its altitude. The formula for it is given below.

Area of a Trapezium = h \( \frac{(a + b)}{2} \)

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Isosceles Trapezium

A trapezium that has equal non-parallel sides and equal base angles is an isosceles trapezium. It is a trapezium with a line of symmetry dividing it. Both the parts of the trapezium look like mirror images of each other.

The formula to calculate the area of an isosceles trapezoid is,

Area of Isosceles Trapezoid = h \( \frac{(a + b)}{2} \)

Parallelogram

Parallelogram ABCD

It is a quadrilateral wherein both pairs of opposite sides are parallel. Also, the lengths of the opposite sides are equal. In the figure shown above, AB is parallel to CD and AD is parallel to BC. Also, AB = CD and AD = BC.

The diagonals bisect each other. As a result, each diagonal divides the parallelogram into two congruent triangles. In the parallelogram ABCD, AC and BD are diagonals. They bisect each other at E. The diagonal AC divides the parallelogram into two congruent triangles ABC and ACD. Likewise, diagonal BD gives two congruent triangles ABD and BCD.

In a parallelogram, the angles are not right angles. The opposite angles are equal and the sum of the angles is 360°.

To find the area of a parallelogram, multiply its base with its height. So, the formula is

Area of Parallelogram= b×h

Properties of Parallelogram

  • Pair of parallel sides are congruent, For example, AB = DC
  • The opposite angles are congruent. For example, D = B
  • If one angle is right, then all are right.
  • The diagonals bisect each other.

Rhombus

Rhombus ABCD

It is a quadrilateral wherein both pairs of opposite sides are parallel. In addition, all sides are equal in length. In other words, a rhombus is a special type of parallelogram whose all sides are equal. To point out, a rhombus looks like a diamond. The angles in a rhombus are congruent to each other. In addition, each diagonal is a perpendicular bisector of the other. To put it differently, they intersect each other at right angles and cut each other in half.

In rhombus ABCD,

  • Side AB is parallel to DC and side BC is parallel to AD
  • All sides are equal to each other i.e. AB=BC=CD=AD
  • Diagonal BD is the perpendicular bisector of diagonal AC and vice versa
  • The diagonals create four triangles, Δ CDA, Δ CBA, Δ DAB and Δ DCB.

Properties of Rhombus

  • Internal angles do not form a 90-degree angle.
  • Sum of the adjacent sides is equal to 180 degrees.
  • The area can be calculated as below
    • If the base and height are given,

Area of Rhombus= b×h

    • When the length of the diagonals are given,

Area of Rhombus= \( \frac{(d_1 × d_2)}{2} \)

where d1 is the diagonal while d2 is the length of the other diagonal.

  • The formula for finding the perimeter is

P = 4s

where ‘ s ‘ is the length of the side.

In summary, the rhombus has all the properties of a parallelogram but all the parallelogram are not the rhombus.

Solved Questions

Q1. Calculate the perimeter of a rhombus whose diagonals are 12 cm and 5 cm long.

  1. 13 cm
  2. 26 cm
  3. 39 cm
  4. 52 cm

Solution: B. The perimeter of a rhombus is 2 √(d1² +d2²) where d1 and d2 are diagonals.
Given, d1=12 cm and d2= 5 cm. Therefore p =
= 2 √(12² + 5²) = 2 √169 = 2 × 13 = 26 cm

Q2. Calculate the area of a trapezoid with bases of 14 cm and 5 cm, and a height of 20 cm.

  1. 180 cm²
  2. 190 cm²
  3. 290 cm²
  4. 170 cm²

Solution: B. Area of a trapezoid is \( \frac{1}{2} \) b1+ b2× h
b1= 14 cm; b2 = 5 cm; h = 20 cm
A = \( \frac{1}{2} \) ( 14 + 5 ) × 20
A = \( \frac{1}{2} \) × 19 × 20
A = 190 cm²

Q. Are all the sides of a rhombus equal?

Ans. Yes, all sides of a rhombus are equal.

Q. Why rhombus is not a regular polygon?

Ans. Rhombus is not a regular polygon because all the angles are not the same. For a polygon to be regular, all the edges and all the angles required to be equal. In fact, among quadrilaterals, only the squares are regular but not a rhombus.

Q. Which rhombus comprises the right angles?

Ans. Indeed, a rhombus can comprise right angles. In fact, a rhombus in which all sides are equal and all angles measure 90 degrees is called a square.

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