Chapter 11: Area Related to Circle

We all love to eat pizzas, especially kids. And a variety of pizzas are available in India which have been modified according to Indian taste and menu. From the Greeks to the Egyptians, from the persians to the Indians, there have been incarnations of pizza served throughout history. Flatebreads, naan and plakountos are all early preparations that could be considered cousins to the modern pizza, but there is not a consensus as to which is first and whether these could even be considered precursors to pizza at all.(Class 10 case study chapter 11 Area related to circle)

Consider two pizzas, both of equal diameter, namely, 12 inches. The first pizza marked (i) has been cut into six equal slices, whereas the second pizza marked (ii) has been cut into eight equal slices.

(A) Find the perimeters of each pizza slices shown in (i and ii).

(B) Find the ratio of areas of each slice of pizza (i) and (ii).

Solution:

(A) Perimeter of aa sector = l + 2r

Where, l is the length of the arc

$l = \dfrac{\theta}{360}\times 2\pi r$

Since, pizza (i) cut into six equal slices.

So angle subtended by each slice at the centre

$\theta_1 = \frac{360}{6}$ = 60°

Similarly, pizza (ii) is cut into 8 equal slices, So angle subtended by each slice at the centre

$\theta_2 = \frac{360}{8}$ = 45°

Also, radius of each slice(r) = 12/2 = 6inches

∴ Perimeter of aa slice of pizza l = Perimeter of a sector

$= l_1 + 2r$

$= \dfrac{\theta_1}{360}\times 2\pi r + 2r$

$= \dfrac{60}{360}\times 2\times \pi \times 6 + 2 \times 6$

$= (2\pi + 12)$ inches

And perimeter of a slice of pizza (ii)

$= l_2 + 2r$

$= \dfrac{\theta_2}{360}\times 2\pi r + 2r$

$= \dfrac{45}{360}\times 2\times \pi \times 6 + 2 \times 6$

$= (1.5\pi + 12)$ inches

(B) Area of one slice of pizza (i)

$= \dfrac{\theta}{360}\times \pi r^2$

$= \dfrac{60}{360}\times \pi (6)^2$

= 6 π

Area of one slice of pizza (ii)

$= \dfrac{\theta}{360}\times \pi r^2$

$= \dfrac{45}{360}\times \pi (6)^2$

$=\dfrac{1}{8}\times \pi \times 36$

= 4.5 π

The ratio of areas of each slice of pizza (i) and (ii)

$= \dfrac{6 \pi}{4.5 \pi}$

= 4 : 3

Question:

The south central zone of Indian Railways has commissioned the longest electrified railway tunnel in the country. According to local media reports, the construction of the 6.6 km long tunnel took 43 months to complete. The electrified railway tunnel is situated between Chelopalli and Rapuru station as a part of the Obulavaripalli-venkatachalam railway line. The tunnel is expected to facilitate freight movement in the region,reducing the distance between Krishnapatnam port and Hinterland areas by nearly 60 km.(Class 10 case study Chapter 11 Area related to circle)

The adjoining figure shows a cross section of the railway tunnel, which is a part of a circle. The radius OA of the circular part is 4 m and ∠AOB = 90°.