Areas of Parallelogram and Rhombuses with Numerical

Key Concepts

After this lesson, students will be able to:

• Area of a Closed figure.

• Recall the formulae for area of Rectangle.

• Find the area formula of a parallelogram.

• Find the area of a rhombus

7.1.1 Area of Planar Region’s 

The part of the plane enclosed by a simple closed figure is called a planner region corresponding to that figure. The magnitude or measure of this planar region is its area. 

Now consider the figures given below: 

Area of figure (X) = Area of figure (P) + Area of figure (Q) 

Similarly,    

Area of figure (Y) = Area of (A)+ Area of (B)+ Area of (C)Area of figure (Y) = Area of (A) + Area of (B) + Area of (C) 

Area of (Z)= Area of (E) +Area of (F) 

7.1.2 Area of Rectangle 

Example 1. 

Area of Rectangle = 8 square units  

              = 4 units x 2 units 

∴Area of rectangle = length x breadth  sq.units 

7.1.3 Finding formulae for the area of a Parallelogram 

A parallelogram is a simple quadrilateral with two pairs of parallel sides. Every rectangle is a parallelogram as well as every rhombus and square. 

Area of a parallelogram is a region covered by a parallelogram in a two-dimensional plane. In geometry, a parallelogram  is a two-dimensional figure with four sides. It is a special case of the quadrilateral, where opposite sides are equal and parallel. The area of a parallelogram is the space enclosed within its four sides. 

7.1.4 Formula of a parallelogram 

To find the area of parallelogram  

We have to compose a rectangle from a parallelogram, first decompose the parallelogram into a right angle and a trapezoid. 

Which formulas does the parallelogram area calculator use? 

Did you notice something? The formula for the area of a parallelogram is pretty much the same as for the rectangle area! Why is it so? Have a look at the picture: a parallelogram can be divided into a trapezoid and a right triangle and rearranged to the rectangle.  

Genius! You made the parallelogram into a rectangle. 

The area of the parallelogram equals the area of the rectangle 

Area of a Rectangle            A =  l  x  w  sq.units 

Area of Parallelogram        A = b  x  h  sq.units  

The formula for the area of a parallelogram is A = bh sq.units 

Key intuition: Every parallelogram can be made into a rectangle, which is why we use the same formula to find the area of a parallelogram and a rectangle. 

Example 2: 

Sol. 

                                                                                          Opposite sides are equal in a parallelogram, then 

Area of parallelogram = l x b                                     Area of parallelogram   A = b x h  

                                  = 10 x 8                                                                   =10 x 8 

∴Area of parallelogram = 80 sq.units                    Area of   parallelogram   A   = 80 sq.units 

So, both areas are equal. 

Example 3 

What is the area of the parallelogram ABCD? 

Solution: 

Step1: Opposite sides of a parallelogram are equal AB = CD = 6 

                                               Base of parallelogram CD = b = 6 units 

                                               Height of parallelogram AE= h=4 units 

Area of given parallelogram A = b x h sq.units 

                                                           =6 x 4 

∴Area of the given parallelogram = 24 sq.units 

7.1.4 Finding the area of a Rhombus  

A rhombus is a simple quadrilateral with all sides equal. The other names are an equilateral quadrilateral or a diamond. 

A rhombus is a parallelogram, as any shape needs to have two pairs of parallel sides to be a parallelogram and the rhombus has them. So, the rhombus is always a parallelogram, but a parallelogram is a rhombus only in a special case – for a parallelogram with four sides of equal length. 

So, the area of a rhombus = The area of a parallelogram = b x h sq.units 

Example 4: 

The pendant at the right is in the shape of a rhombus. A rhombus is a parallelogram with sides of equal length. What is the area of the pendant? 

Solution: Shape of pendant = Rhombus 

Base of the Rhombus b = 3.8 cm 

Base of the height h = 3 cm 

 Area of pendant A = Area of Rhombus  

      A = bh sq.units 

      A = 3.8 x 3 

      A = 11.4cm2 

7.1.5 Practice and & Problem solving   

1. The area of the parallelogram is 132 in2 in2. What is the height of the parallelogram? 

Given that,                                                                                        

Base(b)= 11 in 

Area of parallelogram =132in2

     b.h = 132in2

        11 x h = 132 

11 x h / 11 = 132/11

        h = 12 in 

∴The height of parallelogram is 12 inches. 

2. Micah and Jason made parallelogram-shaped stained-glass windows with the same area. The height of Micah’s window is 9 inches height, and its base is 10 inches. The height of Jason’s window is 6 inches. What is the base of Jason’s window? 

Case 1. 

Micah measurements for window 

Height(h)= 9 in 

base (b)= 10 in 

Then, 

Area of the window = b.h 

= 9 x 10 

Area of the window = 90.in2

Case 2. 

Jason measurements for window          

height(h)= 6 in 

base(b) = ? 

Area of Jason window = Area of Micah window 

Area of Jason window = 90in2

bh  = 90  

b x 6 = 90 

b x 6 / 6 = 906

b=15 in 

Case 3. The area of the rhombus is 52 m2. What is the base of the rhombus? 

Given that, 

height(h)= 6.5 m 

base(b)=? 

Area of Rhombus = 52m2

b.h = 52 

b x 6.5 = 52 

b x 6.5 / 6.5 =  52/6.5

   b = 8 m 

∴The base of the Rhombus is 8 m 

4. A rectangle has a length of 8 m and a width of 4.5 m. A parallelogram has a length of 6 m. The area of the parallelogram is twice the area of the rectangle. What is the height of the parallelogram?  

Given that, 

Length of rectangle (l)= 8 m 

Breadth of Rectangle (w)= 4.5 m 

 Area of rectangle A  = l. w  sq.units 

                                       =  8 x 4.5 

 Area of rectangle A = 36 m2

Length of Parallelogram  = 6m 

Height of parallelogram = ? 

We know that , 

Area of parallelogram = 2(area of rectangle) 

b.h = 2 x 36  

6 x  h =   72 

6 x h / 6 = 72/6

h    =  12 m 

∴The height of parallelogram is 12 m 

5. Hilary made an origami dog. What is the area of the parallelogram that is highlighted in the origami figure?  

 

Given that, 

Height(h) = 2.36 cm 

    base(b) =  4 cm 

Area of parallelogram  A = bh    

       =  2.36 x 4 

       =  9.44cm2

∴The area of parallelogram is 9.44cm2

6. A type of origami paper comes in 15 cm by 15 cm square sheets. Hilary used two sheets to make the origami dog. What is the total area of the origami paper that Hilary used to make the dog?  

Given that,  

Side of a square = 15 cm 

Area of square =  s x s 

  =  15 x 15  

    =  225cm2

The dog is made from two square papers 

 so, 

2(area of square) = 2 x 225 

       = 450cm2

∴The paper needed to make the dog is 450cm2.

7. A rectangle and a parallelogram have the same base and the same height. How are their areas related? 

Provide an example to justify your answer. 

Let take a rectangle and parallelogram with same base and height 

Length of rectangle be 4 m 

Breadth of rectangle be 2 m 

Area of rectangle = l.w sq.units  

       = 4×2 

Area of rectangle = 8m2

Length of parallelogram be 4 m  

Height of parallel gram be 2 m 

Area of parallelogram = b.h sq.units 

  =  4 x 2 

Area of parallelogram =  8m2

∴ If the rectangle and parallelogram have the same height and base then their areas are equal 

8. Soshi’s rhombus has a base of 12 in and a height of 10 in. Jack’s rhombus has base and height measured that are double those of Soshi’s rhombus. Compare the area of Jack’s rhombus to the area of Soshi’s rhombus. Explain. 

Given that,  

Step1.Measurments of Sochi’s rhombus 

height (h) = 10 in 

base (b) = 12 in 

Area of Sochi’s rhombus = bh 

       = 10 x 12  

Area of Sochi’s rhombus = 120in2

Step2. 

Measurements of Jack’s rhombus are twice the measurements of  

Sochi’s rhombus 

height(h) = 2 x 10 = 20 in 

base(b)  = 2 x 12 = 24  in 

Area of Jack’s rhombus = bh 

     = 20 x 24 

Area of Jack’s rhombus = 480in2

Step 3. The ratio of area of Sochi’s rhombus and Jack’s rhombus = 120 : 480 

1: 4 

∴Area of Jack’s rhombus is 4 times greater than that of Sochi’s.

Exercise:

1. The measurements of some rectangles are given in the table below. However, they are incomplete. Find the missing information. Length Breadth Area 20 cm 14 cm 12 cm 48cm2 15 cm 150 cm2

2. 1. In a parallelogram ABCD, AB = 10 cm and DE = 4 cm. Find L The area of ABCD.

ii. The length of BF, if AD = 6 cm

3. The height of a parallelogram is one third of its base. If the area of the parallelogram is 192m2 find its height and base?

4. In a parallelogram, the base and the height are in the ratio of 5:2.1f the area of the parallelogram is 360m2 find its base and height?

5. A square and a parallelogram have the same area. If a side of the square is 40 m and the height of the parallelogram is 20 m, find the base of the parallelogram?

What have we learned?

• Area of a closed figure.

• Recall the formulae for area of rectangle.

• Find the area formula of a parallelogram.

• Find the area of a rhombus