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Area and Perimeter of Composite Figures

Problem 1

The figure below is made of two identical rectangles and two identical quadrants. Find the perimeter of the shaded part.

find perimeter of shaded part
Breadth of rectangle
= 1 cm + 0.6 cm = 1.6 cm




Perimeter contributed by top rectangle  =  0.6 + 2.4 + 1.6
   =  4.6 cm
Perimeter contributed by bottom rectangle  =  0.6 + 2.4 + 1.6
   =  4.6 cm
Perimeter contributed by left quadrant  =  1
42
× 21 × 3.14 × 1
   =  1.57 cm
Perimeter contributed by right quadrant  =  1
42
× 21 × 3.14 × 1
   =  1.57 cm

Perimeter of shaded part  =  Perimeter contributed by top rectangle
  + Perimeter contributed by left quadrant
  + Perimeter contributed by bottom rectangle
  + Perimeter contributed by right quadrant
   =  4.6 + 1.57 + 4.6 + 1.57
   =  12.34 cm

Problem 2

The figure below shows two identical quadrants in two identical squares and a rectangle joining them to form a butterfly. Find the perimeter of the shaded butterfly.

find perimeter of shaded butterfly

Perimeter contributed by 1 square  =  2 + 2
   =  4 cm
Perimeter contributed by 1 quadrant  =  1
421
× 21 × 3.14 × 21
   =  3.14 cm
Perimeter contributed by rectangle  =  0.2 + 2 + 0.2 + 2
   =  4.4 cm

Perimeter of shaded butterfly = 2 × Perimeter contributed by 1 square
  + 2 × Perimeter contributed by 1 quadrant
  + Perimeter contributed by rectangle
  = (2 × 4) + (2 × 3.14) + 4.4
  = 8 + 6.28 + 4.4
  = 18.68 cm

Problem 3

Maddie joined 3 semicircles and a rectangle to form the shape of an umbrella. Find the area of the shaded umbrella.

find area of shaded umbrella

Area of big semicircle  =  
1
21
 × 2211
71
 × 71 × 7
   =  77 cm2
Radius of medium semicircle
= 2 cm + 3 cm = 5 cm




Area of medium semicircle  =  1
2
 × 3.14 × 5 × 5
   =  39.25 cm2

Area of small semicircle  =  1
2
 × 3.14 × 3 × 3
   =  14.13 cm2

Area of rectangle  =  10 × 2
   =  20 cm2

Area of shaded umbrella  =  Area of big semicircle
  + Area of rectangle
  + Area of medium semicircle
  − Area of small semicircle
   =  77 + 20 + 39.25 − 14.13
   =  122.12 cm2

Problem 4

Find the area of the shaded heart formed by 2 semicircles and a square.

find area of shaded heart
Radius of semicircle
= 12 m ÷ 2
= 6 m
Area of 2 semicircles  =  Area of 1 circle
   =  3.14 × 6 × 6
   =  113.04 m2

Area of square  =  12 × 12
   =  144 m2

Area of shaded heart  =  Area of 2 semicircles
  + Area of square
   =  113.04 + 144
   =  257.04 m2

Problem 5

Find the area of the figure below formed by 2 identical three-quarter circles, 2 isosceles right-angled triangles and a square.

find area of shaded figure
Radius of circle
= 8 cm − 4 cm
= 4 cm




Area of 1 three-quarter circle  =  3
41
 × 3.14 × 41 × 4
   =  37.68 cm2

Height of right isosceles triangle
= Base of right isosceles triangle
= 4 + 10 = 14 cm





Area of big isosceles right triangle  =  1
21
 × 147 × 14
   =  98 cm2

Area of small isosceles right triangle  =  1
21
 × 84 × 8
   =  32 cm2

Area of square  =  8 × 8
   =  64 cm2

Area of shaded figure  =  2 × Area of 1 three-quarter circle
  + Area of big isosceles triangle
  + Area of small isosceles triangle
  + Area of square
   =  (2 × 37.68) + 98 + 32 + 64
   =  75.36 + 98 + 32 + 64
   =  269.36 cm2

Problem 6

Find the area and perimeter of the shaded figure below.

find area and perimeter of shaded figure
Diameter of semicircle
= 2.8 + 1.4 + 2.8 = 7 cm


PERIMETER

Perimeter contributed by 1 semicircle  =  1
21
 × 2211
71
 × 71
   =  11 cm

Perimeter contributed by 1 quadrant  =  1
421
 × 21 × 2211
71
 × 2.80.4
   =  4.4 cm

Perimeter contributed by rectangle  =  3.5 + 3.5
   =  7 cm

Perimeter of shaded figure  =  2 × Perimeter contributed by 1 semicircle
  + 4 × Perimeter contributed by 1 quadrant
  + Perimeter contributed by rectangle
   =  (2 × 11) + (4 × 4.4) + 7
   =  22 + 17.6 + 7
   =  46.6 cm

Radius of semicircle
= 7 cm ÷ 2
= 3.5 cm



AREA

Area of 1 semicircle  =  1
21
 × 2211
71
 × 3.50.5 × 3.5
   =  19.25 cm2

Area of 1 square  =  2.8 × 2.8
   =  7.84 cm2

Area of 1 quadrant  =   1
41
 ×  22
71
 × 2.80.7 × 2.80.4
   =  6.16 cm2
Length of rectangle
= 3.5 + 2.8 + 2.8 = 9.1 cm



Area of rectangle  =  9.1 × 1.4
   =  12.74 cm2

Area of shaded shape  =  2 × Area of 1 semicircle
  + 4 × (Area of 1 square − Area of 1 quadrant)
  + Area of rectangle
   =  (2 × 19.25) + (4 × (7.84 − 6.16))+ 12.74
   =  38.5 + 6.72 + 12.74
   =  57.96 cm2