Session 7: Area of Composite Shapes – Definition, Formula, Interactives and Examples

What is Area?

The area is the amount of surface enclosed by a closed two-dimensional figure. It is measured by the number of unit squares it takes to cover a two-dimensional shape. For example, if you count the small squares, you will find there are 15 of them. Therefore, the area is 3⋅5 or 15 unit².

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What are Composite Shapes?

A composite shape or a composite figure is a two-dimensional figure made up of basic two-dimensional shapes such as triangles, rectangles, circles, semi-circles, etc.

 

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Finding the Area of Composite Shapes

Find the individual areas of each piece of the composite shapes. The area of the composite shape will be the sum of the individual areas.

1) Find the perimeter and area of the composite shape below.

The given shape has been broken into 2 rectangles (although it could have been broken up differently).

First, find all the missing side lengths.

Now we can find the perimeter by finding the sum of all the side lengths.

Perimeter=10+5+6+7+4+12=44 cm

Now, we find the area of each of the two pieces and then the total area.

Area of Rectangle #1, A₁=bℎ=5×6=30 cm²

Area of Rectangle #2, A₂=bℎ=12×4=48 cm²

Total Area, A₁+A₂=30+48=78 cm²

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Area of Composite Shapes – Examples

Example 1

Find the area of the shape shown below.

The given shape has been broken down into one rectangle and one square (although it could have been broken up differently).

To find the area of the composite figure, first, we find the individual areas of each piece of the composite figure and then the sum of the individual areas.

Area of Rectangle, A₁=lb=12×8=96 units²

Area of Square, A₂=s²=4×4=16 units²

Total area, A₁+A₂=96+16=112 units²

Example 2

From a square sheet of paper of side 14 cm, a strip of width 3 cm is removed along the border. Find the area of the removed strip of paper.

Side of the outer square sheet = 14 cm

Width of the strip = 3 cm

Side of the inner square sheet =14−2(3)=14−6=8 cm

Area of the outer square =14×14=196 cm²

Area of the inner square =8×8=64 cm²

Area of the removed strip = Area of the outer square – Area of the inner square

 =196−64
=132 cm²

Example 3

A rectangular park 60 m long and 40 m wide has a 2 m wide path along the sides of the park. Find the area of the path and the cost of its construction at the rate of ₹10/ m².

Here, the shaded region represents the path and EFGH represents the field.

The length of the field, EF=60 m.

The breadth of the field, EH=40 m.

The width of the path=2 m.  

The length of the field including the path, AB=(60+2×2) m=64 m.

The breadth of the field including the path, AD=(40+2×2) m=44 m.

Area of the field including the path = Area of ABCD                                                          

Area of the field excluding the path = Area of EFGH                                                           

Area of path = Area of ABCD − Area of EFGH

=2816−2400
=416 m²

Rate of constructing path=₹10/m².

Cost of constructing 1 m2 path =₹10.

Cost of constructing 416 m² path=10×416=₹4160.

Example 4

A rectangular garden 200 m × 170 m has two paths, each 5 m wide, running in the middle of it, one parallel to the length and the other parallel to breadth as shown in the figure. Find the cost of gravelling the path at ₹8.50/ m².

Here area of the path parallel to the length of the garden = Area of EFGH=EF×EH

                                                                                                 =5×200=1000 m²

Area of the path parallel to the breadth of the garden = Area of IJKL=IJ×IL

                                                                                           =5×170=850 m²

It is clear that the total area of the path is the area of the orange portion i.e. the sum of the area of the rectangle EFGH and area of the rectangle IJKL. While doing this, the area of the square ABCD is taken twice, which needs to be subtracted.

∴ The total area of the paths = Area of  EFGH + Area of IJKL − Area of ABCD                                               

Rate of gravelling =₹8.50/ m2

Cost of gravelling 1 m²=₹8.50

Cost of gravelling 1825 m²=8.50×1825

                                              =₹15512.50

Remember this!

• The area is the amount of surface enclosed by a closed two-dimensional figure.
• A composite shape is a two-dimensional figure made up of basic two-dimensional shapes such as triangles, rectangles, circles, semi-circles, etc.
• The area of a composite shape is the sum of the individual areas of each piece of the composite shape.