1. Composite Shapes Circular Parts
with
Circular Parts
© T Madas

2. © T Madas

3. π π A semicircle has a radius of 9 cm. Calculate its area
Calculate its perimeter
127 cm2 π
A =
x r 2 c π
A =
x 92 c
9 cm
A ≈
254
cm2
© T Madas

4. π π A semicircle has a radius of 9 cm. Calculate its area
Calculate its perimeter
28.25 cm π
C =
2 x
x r c π
C =
2 x
x 9 c
9 cm
C ≈
56.5 cm
P =
28.25
+ 18
= cm
© T Madas

5. © T Madas

6. π π A quarter-circle has a radius of 16 cm. Calculate its area
Calculate its perimeter π
201 cm2
A =
x r 2 c π
A =
x 182 c
16 cm
A ≈
804
cm2
© T Madas

7. π π A quarter-circle has a radius of 16 cm. Calculate its area
Calculate its perimeter
25.1 cm π
C =
2 x
x r c π
C =
2 x
x 16 c
16 cm
C ≈
100.53 cm
P =
25.1
+ 16
+ 16
= 57.1 cm
© T Madas

8. © T Madas

9. Calculate the perimeter and area of the following composite shape.1 2
x π
P = 10
+ 8
+ 10 + x 2
x 4
4 cm
x π
P = 28
+ 4
10 cm
P ≈
40.6 cm
8 cm
x π
C = 2
x r
© T Madas

10. π π Calculate the perimeter and area of the following composite shape.1 2
x π
P = 10
+ 8
+ 10 + x 2
x 4
4 cm
x π
P = 28
+ 4 A1 cm
10 cm
P ≈
40.6 1 2 π
A = 10
x 8 + x
x 4 2
8 cm
x π
A = 80
+ 8
x π
C = 2
x r
A ≈
105
cm2 π
A =
x r 2
© T Madas

11. © T Madas

12. Calculate the perimeter & area of the grey region below.
6 cm
© T Madas

13. π Calculate the perimeter & area of the grey region below.
The perimeter of the grey area is equal to …
… the circumference of a circle of radius …
… 3 cm
6 cm
x π
P = 2
x 3
P ≈
18.8 cm
x π
C = 2
x r π
A =
x r 2
© T Madas

14. π π Calculate the perimeter & area of the grey region below.
The area of the grey area is equal to …
… the area of a square with side 6 cm …
… less …
… the area of a circle of radius 3 cm
6 cm π
A = 6
x 6 –
x 3 2
x π
x π
C = 2
x r
A = 36
– 9 π
A =
x r 2
A ≈
7.73
cm2
© T Madas

15. © T Madas

16. π π π π [ ] x Calculate the perimeter & area of the following shape
The perimeter is equal to …
… the circumference of a semi-circle of radius 20 cm …
… plus …
… the circumference of a circle of radius 10 cm …
10 cm
40 cm
x π
[ ] x 1 2
x π
P = 2
x 20
+ 2
x 10 π π
P = 20
+ 20
x π
C = 2
x r π
P = 40 π
A =
x r 2
P ≈
125.7 cm
© T Madas

17. π π π π π [ ] x Calculate the perimeter & area of the following shape
The total area is equal to …
… the area of a semi-circle of radius 20 cm …
… plus …
… the area of a circle of radius 10 cm …
10 cm
40 cm π
[ ] x 1 2
+ π
A =
x 20 2
x 10 2 π π
A =
200
+ 100
x π
C = 2
x r π
A =
300 π
A =
x r 2
A ≈
942
cm2
© T Madas

18. © T Madas

19. Calculate to 2 decimal places: the perimeter of the pond.
The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m.
Calculate to 2 decimal places:
the perimeter of the pond.
the area of the pond
4 m
4 m
© T Madas

20. The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m.
Calculate to 2 decimal places:
the perimeter of the pond.
the area of the pond C = 2 x π x r c 4
4 m
6.28 ? C = 2 x π x 4 c 4 C ≈
25.13 m
Each curved edge is ¼ of the circumference of a full circle. 4
6.28 ?
25.13 ÷ 4 ≈
6.28 m 4 P = 4 x 4 + 2 x
6.28 ≈
28.57 m
© T Madas

21. The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m.
Calculate to 2 decimal places:
the perimeter of the pond.
the area of the pond A = π x r 2 c
16 m2 A = π x 4 2 c
12.57 m2 A = π x 16 c
4 m
4 m A ≈
50.27 m2
12.57 m2
16 m2
50.27 ÷ 4 ≈
12.57 m2
Area of a Quarter Circle P = 2 x 16 + 2 x
12.57 ≈
57.14 m2
© T Madas

22. © T Madas

23. π π [ ]x 4 Calculate the perimeter & area of the following shape:
The perimeter is equal to …
… the circumference of …
… 8 semi-circles of radius 4 m …
… or …
… 4 circles of radius 4 m …
4 m
16 m
x π
P =
[ ]x 4 2
x 4 π
P = 32
P ≈
100.5 m
x π
C = 2
x r π
A =
x r 2
© T Madas

24. π π [ ]x 4 Calculate the perimeter & area of the following shape:
The total area is equal to …
… the area of a square with side length of 16 m …
… plus …
… the area of 4 circles of radius 4 m …
4 m
16 m
[ ]x 4 π
A = 16
x 16 +
x 4 2
x π
A =
256
+ 64
x π
C = 2
x r
A ≈
457 m2 π
A =
x r 2
© T Madas

25. © T Madas

26. π π π π [ ] x Calculate the perimeter & area of the following shape:
The perimeter is equal to …
… the circumference of a semi-circle of radius 14 cm …
… plus …
… the circumference of a circle of radius 7 cm …
7 cm
28 cm
x π
[ ] x 1 2
x π
P = 2
x 14
+ 2
x 7 π π
x π
P = 14
+ 14
C = 2
x r π
P = 28 π
A =
x r 2
P ≈
88.0 cm
© T Madas

27. π π π π π [ ] x Calculate the perimeter & area of the following shape:
The total area is equal to …
… the area of a semi-circle of radius 14 cm …
… less…
… the area of a circle of radius 7 cm …
7 cm
28 cm π
[ ] x 1 2
– π
A =
x 14 2
x 7 2 π π
x π
A = 98
– 49
C = 2
x r π
A = 49 π
A =
x r 2
A ≈
154
cm2
© T Madas

28. Harder Problems
© T Madas

29. Break my Heart...
© T Madas

30. Find the area of the heart in terms of a
Area of square:
2 semi-circles = circle
Note that the circle’s radius is: a a
© T Madas

31. © T Madas

32. © T Madas

33. Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them.
© T Madas

34. Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them. 1
solution
© T Madas

35. Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them.
Ac = π
r 2 c
Ac = π
x 12 c
Ac = π
the total area is 1
the area of this composite consists of a circle of unit radius
plus a 1 by 2 rectangle
© T Madas

36. Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them.
Ac = π
r 2 c
Ac = π
x 12 c
Ac = π
the total area is
π + 2 1
the area of this composite consists of a circle of unit radius
plus a 1 by 2 rectangle
© T Madas

37. © T Madas

38. The grey area...
© T Madas

39. Find the area enclosed by the 4 circles in terms of a
Area of the square: a
4 quarter-circles = circle
© T Madas

40. © T Madas

41. ...flower petals...
© T Madas

42. Find the exact area of the orange “petal”
Billy wants a hint ... a
© T Madas

43. © T Madas

44. Find the exact area of the orange “petal”
one of the blue regions:
area of the square
less the area of the quarter circle
both blue regions a
The area of the “petal” is given by the area of the square less the area of the two blue regions:
© T Madas

45. © T Madas

46. Vase equals Square
© T Madas

47. ? Vase equals Square Look at this vase shaped object
It consists of 6 identical arcs
Each arc is a quarter circle
If the quarter circles to which these arcs correspond have radius a, find the area of this object a a ?
© T Madas

48. ? Vase equals Square Look at this vase shaped object
It consists of 6 identical arcs
Each arc is a quarter circle
If the quarter circles to which these arcs correspond have radius a, find the area of this object a a ?
1st hint
2nd hint
© T Madas

49. Vase equals Square a a
1st hint
2nd hint
© T Madas

50. Vase equals Square 4a 2 No complex calculations needed ! 2a
The area of this object is equal to the area of the square on the right.
No complex calculations needed !
Vase equals Square a a
4a 2 2a
1st hint
2nd hint
© T Madas

51. another approach ...
© T Madas

52. © T Madas

53. Yin Yang
© T Madas

54. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections a
© T Madas

55. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections a
© T Madas

56. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections
A r e a
working with the green section: a
These semicircles both have a radius of
© T Madas

57. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections
A r e a
working with the green section:
The area of the green section of the Yin Yang is equal to the area of a semicircle a
These semicircles both have a radius of
© T Madas

58. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections
A r e a
working with the green section: a
© T Madas

59. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections
P e r i m e t e r
working with the green section;
the required perimeter is given by:
the circumference of a circle of diameter
plus the circumference of a semicircle of diameter a
© T Madas

60. In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections
P e r i m e t e r a
© T Madas

61. © T Madas

62. ?
Spiral Galaxies
Comets
Yin Yang
Marbles
© T Madas

63. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections. a
© T Madas

64. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections. a
© T Madas

65. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
The blue sections are congruent
Area of the blue sections a
This semicircle has a radius of
© T Madas

66. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
Area of the blue sections a
This semicircle has a radius of
This semicircle has a radius of
© T Madas

67. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
Area of the blue sections a
© T Madas

68. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
Area of the orange section
This is best found by subtracting the areas of the two blue sections we just found from the whole circle a
© T Madas

69. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
perimeter of a blue section a
© T Madas

70. Look at the shape below, consisting of three sections.
Calculate the area and perimeter of these sections.
perimeter of a blue section
perimeter of the orange section a
© T Madas

71. The generalisations of the Yin Yang shape:
The circle in every case :
is divided by curved lines of equal lengths
the resulting regions have equal perimeters
the resulting regions have equal areas
© T Madas

72. © T Madas

73. ... another grey area...
© T Madas

74. Find the grey area enclosed by the 3 circles in terms of a
The grey area is equal to the area of an equilateral triangle of side 2a less a semicircle of radius a a
Area of Triangle: 2a
60° 2a
© T Madas

75. Find the grey area enclosed by the 3 circles in terms of a
The grey area is equal to the area of an equilateral triangle of side 2a less a semicircle of radius a a
Area of Triangle:
Area of semicircle:
The grey area:
© T Madas

76. © T Madas

77. Broomsticks and Elastic Bands Broomsticks and Elastic Bands
© T Madas

78. How long is the elastic band in terms of a ?
Three cylindrical broomsticks each of radius a are held together by an elastic band.
How long is the elastic band in terms of a ?
Solution
all the distances between the centres of the circles are 2a , so we have an equilateral triangle at the centre.
60° a
© T Madas

79. How long is the elastic band in terms of a ?
Three cylindrical broomsticks each of radius a are held together by an elastic band.
How long is the elastic band in terms of a ?
Solution
120°
Draw radii as shown towards the elastic band.
The radii must be at right angles at the points of contact with the elastic band. (tangent – radius)
We can now work another useful angle
60° a
© T Madas

80. How long is the elastic band in terms of a ?
Three cylindrical broomsticks each of radius a are held together by an elastic band.
How long is the elastic band in terms of a ?
Solution
120°
We can now calculate some lengths.
Each straight piece (not in contact with the circles) has length 2a
Each arc corresponds to one third of a circle
Finally do all the adding 2a
60° a
© T Madas

81. How long is the elastic band in terms of a ?
Three cylindrical broomsticks each of radius a are held together by an elastic band.
How long is the elastic band in terms of a ?
Solution
120° 2a
60° a
© T Madas

82. © T Madas

83. Annulus Problem 1
© T Madas

84. 3 4 5
Three concentric circles have radii of 3, 4 and 5 units, as shown opposite.
What percentage of the largest circle is shaded?
Annulus:
So:
© T Madas

85. © T Madas

86. Train Sets
© T Madas

87. D
Marcus has two circular railway lines, one with radius of 1.5 metres and the other with radius of 2 metres.
He runs an engine clockwise round each track at the same speed from the start line of the diagram.
Where would the engine on the outer track be, out of A, B, C or D when the engine on the inner track has made 11 complete circuits? 2
1.5 C A B
© T Madas

88. D C A B 2 A circuit on the inner track: A circuit on the outer track:
11 complete circuits on the inner track:
Both engines travel at the same speed, so the engine on the outer track must also cover a distance of 33π, with each circuit in the outer track being 4π 2
1.5 C A B
The engine on the outer track will be at point B when the engine on the inner track has completed 11 circuits.
© T Madas

89. © T Madas