QUESTION 5
Easy
Find the perimeters of the following shapes(taking the arcs to be quarter or half or three-quarters of a circle, as appropriate):
SOLUTION
1
(i)
Diameter \(= 60\,m\)
Radius, \(r = \frac{60}{2} = 30\,m\)
=
\(\begin{aligned} Perimeter &= 2(straight\, side)\, +\, Circumference\, of\, one \,circle \\ &= 2(80) + 2πr \\ &= 160 + 2 × \frac{22}{7} × 30 \\ &= 348.57\,m\end{aligned}\)
The perimeter is 348.57 m.
Radius, \(r = \frac{60}{2} = 30\,m\)
=
\(\begin{aligned} Perimeter &= 2(straight\, side)\, +\, Circumference\, of\, one \,circle \\ &= 2(80) + 2πr \\ &= 160 + 2 × \frac{22}{7} × 30 \\ &= 348.57\,m\end{aligned}\)
The perimeter is 348.57 m.
2
(ii)
Outer semicircle:
Diameter \(= 12 \,cm\)
Radius \(= 6\,cm\)
Arc length \(= πr\,= \frac{22}{7} × 6 = 18.86\,cm\)
Inner semicircle:
Diameter \(= 8\,cm\)
Radius \(= 4\,cm\)
Arc length \(= πr\,= \frac{22}{7} × 4 = 12.57\,cm\)
Straight ends:
Outer diameter = 12 cm
Inner diameter = 8 cm
Difference = 12 − 8 = 4 cm
There are two equal ends so each end is 2 cm.
Total straight length = 4 cm
Total perimeter = 18.86 + 12.57 + 4 = 35.43 cm
The perimeter is 35.43 cm.
Diameter \(= 12 \,cm\)
Radius \(= 6\,cm\)
Arc length \(= πr\,= \frac{22}{7} × 6 = 18.86\,cm\)
Inner semicircle:
Diameter \(= 8\,cm\)
Radius \(= 4\,cm\)
Arc length \(= πr\,= \frac{22}{7} × 4 = 12.57\,cm\)
Straight ends:
Outer diameter = 12 cm
Inner diameter = 8 cm
Difference = 12 − 8 = 4 cm
There are two equal ends so each end is 2 cm.
Total straight length = 4 cm
Total perimeter = 18.86 + 12.57 + 4 = 35.43 cm
The perimeter is 35.43 cm.
3
(iii)
Diameter \(= 10\,cm\)
Radius \(= 5\,cm\)
Perimeter of the 4 semicircles \(=4 × πr\,=4 × \frac{22}{7} × 5\,=\,\frac{440}{7}\,= 62.86\,cm\)
The perimeter is 62.86 cm.
Radius \(= 5\,cm\)
Perimeter of the 4 semicircles \(=4 × πr\,=4 × \frac{22}{7} × 5\,=\,\frac{440}{7}\,= 62.86\,cm\)
The perimeter is 62.86 cm.
4
(iv)
Diameter \(= 12\,cm\)
Radius \(= 6\,cm\)
Perimeter of the three semicircles \(= 3 × πr\,= 3 × \frac{22}{7} × 6\,= \frac{396}{7}\,= 56.57\,cm
The perimeter is 56.57 cm.
Radius \(= 6\,cm\)
Perimeter of the three semicircles \(= 3 × πr\,= 3 × \frac{22}{7} × 6\,= \frac{396}{7}\,= 56.57\,cm
The perimeter is 56.57 cm.
5
(v)
Concept Note
(i) The shape consists of two straight lines(top and bottom), each 80 m and two semicircles. each having diameter = 60 m. The two semicircles make a complete circle. So
Perimeter = 2(straight side) + Circumference of one circle
(ii) The figure is a semi-circular ring. The perimeter consists of outer semicircle, inner semicircle, and two straight ends.
(iii)The figure is made from four semicircles. The dotted square has side 10 cm. So each semicircle has diameter 10 cm.
(iv) The figure is formed by 3 equal circles touching each other. Only the outer semicircle arcs form the perimeter. So the perimeter is 3 semicircles.