Diameter $$= 17\frac {1}{2} cm = 17.5  cm =>$$ radius $$= 8.75  cm$$

Area of a circle of radius r $$= \pi {r}^{2}$$

Diameter $$= 21.6 \ \text{m}$$
Radius $$= 10.8\ \text{ m}$$

Given that,

Radius of window $$=28\,cm=0.28\,m$$

Let $$A$$ be the area of each circular window

  $$A=\pi {{r}^{2}}\\=\dfrac{22}{7}\times {{\left( \dfrac{28}{100} \right)}^{2}}\\=\dfrac{22\times 28\times 28}{7\times 100\times 100}\\=0.2464{{m}^{2}}$$

Let $$A'$$ be the area of 4 circular windows

$$A'=4\pi {{r}^{2}}\\=4\times 0.2464\\=0.9856\,{{m}^{2}}$$   

The area of the circle $$=12474\,c{{m}^{2}}$$

We know that the area of the circle $$=\pi {{r}^{2}}$$

$$\dfrac{22}{7} \times {{r}^{2}}=12474$$

$$r^2 =12474\times \dfrac{7}{22}$$

$${{r}^{2}}=3969$$

$$r=63\,cm$$

We know that the circumference of the circle

$$=2\pi r$$

$$=2\times \dfrac{22}{7}  \times 63$$

$$=396\,cm$$

$${\textbf{Step -1: Calculating the area covered in one rotation}}{\text{.}}$$

                 $${\text{To find how much area will be covered in one rotation,}}$$

                $${\text{we have to find the curved surface area of cylindrical roller}}{\text{.}}$$  

                 $${\text{Given that Diameter, D}} = 1.4{\text{ m}}{\text{.}}$$

                 $${\text{and length h}} = 2{\text{m}}{\text{.}}$$

                 $${\text{so, radius r}} = \dfrac{{\text{D}}}{2} = \dfrac{{1.4}}{2} = 0.7{\text{m}}{\text{.}}$$

                 $${\text{As per formula, curved surface area}} = 2\pi {\text{rh}}{\text{}}$$

                 $$= 2 \times \dfrac{{22}}{7} \times 0.7 \times 2$$

                 $$= 8.8{{\text{m}}^2}.$$

$${\textbf{Step -2: Calculating the area covered in 5 revolutions}}{\text{.}}$$

                 $${\text{So in 1 revolution total 8}}{\text{.8 }}{{\text{m}}^2}{\text{ area is covered}}{\text{.}}$$

                 $$\therefore {\text{ in 5 revolutions total area covered is 8}}{\text{.8}} \times {\text{5 }}{{\text{m}}^2}.$$

                 $$= 44{\text{ }}{{\text{m}}^2}.$$

$${\textbf{Hence, It will cover total 44 }}{{\textbf{m}}^2}{\textbf{ in 5 revolutions}}{\text{.}}$$

Hence, the area of the path $$= \pi \left( { 112 }^{ 2 }-{ 105 }^{ 2 } \right) = \frac { 22 }{ 7 } \left( 12544-11025 \right) = \frac {33418}{7} = 4774   {m}^{2}$$

Area of the region between two concentric circles with radius of outer circle R, and inner circle 

  $$r= \pi\left( { R }^{ 2 }-{ r }^{ 2 } \right)$$

So $$R = 105$$ $$m$$ & $$r = 100$$ $$m$$ .

Hence, the area of the path $$= \pi \left( { 105 }^{ 2 }-{ 100}^{ 2 } \right) = \cfrac { 22 }{ 7 } \left( 11025-10000 \right) = \cfrac {22550}{7} = 3221 \cfrac{3}{7}  {m}^{2}$$

Area of the region between two concentric circles with radius of outer circle R, and inner circle r $$= \pi \left( { R }^{ 2 }-{ r }^{ 2 } \right)$$

Hence, the area of the circular track $$= \pi \left( { 70 }^{ 2 }-{ 63 }^{ 2 } \right) = \frac { 22 }{ 7 } \left( 4900-3969 \right) = 2926  {m}^{2}$$

Area of the region between two
concentric circles with radius of outer circle R, and inner circle r $$= \pi \left( { R }^{ 2 }-{ r }^{ 2 } \right)$$

Let us assume $$R = 4.5$$ $$m$$ & $$r = 2.5$$ $$m$$ .

Area of the region between two concentric circles 

with radius of outer circle R, and inner circle r $$= \pi \left( { R }^{ 2 }-{ r }^{ 2 } \right)$$

Hence, the area of the region formed by two
concentric circles whose radii are 4.5  m and 2.5 m $$= \pi \left( { 4.5 }^{ 2 }-{ 2.5 }^{ 2 } \right) = \frac { 22 }{ 7 } \left(20.25-6.25\right) = 44   {m}^{2}$$

The area of the circular sheet with $$4\;{\rm{cm}}$$ radius is,

$$A = \pi {r^2}$$

$$= 16\pi$$

The area of the circular sheet with $$3\;{\rm{cm}}$$ radius is,

$$A = \pi {r^2}$$

$$= 9\pi$$

Since the circle with radius $$3\;{\rm{cm}}$$ is removed, then the remaining area is,

$$A = 16\pi  - 9\pi$$

$$= 7\pi$$

$$= 7 \times 3.14$$

$$= 21.98\;{\rm{c}}{{\rm{m}}^2}$$

(a) The radius is given as $$14\ {cm}$$ .

The circumference of the circle is,

$$C = 2\pi r$$

$$= 2 \times \dfrac{{22}}{7} \times 14$$

$$= 88\ {cm}$$

(b)

The radius is given as $$28\ {cm}$$

The circumference of the circle is,

$$C = 2\pi r$$

$$= 2 \times \dfrac{{22}}{7} \times 28$$

$$= 176\ {cm}$$

(c)

The radius is given as $$21\ {cm}$$

The circumference of the circle is,

$$C = 2\pi r$$

$$= 2 \times \dfrac{{22}}{7} \times 21$$

$$= 132\ {cm}$$

Given: The circumference of the circle is $$42$$ cm.

Let the radius and diameter of the circle are  $$r$$ and $$d$$ , respectively.

We know that, 

Circumference $$= 2\pi r$$

$$\therefore 2\pi r=42$$

$$\Rightarrow 2\left ( \dfrac{22}{7} \right ) r=42$$

  $$\Rightarrow r = \dfrac{147}{22} \approx 6.68$$ cm

Since, the diameter of the circle is twice of it radius.

$$\therefore d=2r$$

$$\Rightarrow d =2\left ( \dfrac{147}{22} \right ) = \dfrac{147}{11} \approx 13.36$$

Thus, the diameter of the given circle is $$\dfrac{147}{11}$$ cm or $$13.36$$ (approx) cm.

The circumference is only calculated in case of circle. So, the circumference of the circle is $$448$$ .

Let the radius of the circle is $$r$$ .

Write the formula to calculate circumference of circle.

Circumference $$=2\pi r$$

Substitute the values in the above equation.

$$2\pi r=448$$

$$2\left ( \frac{22}{7} \right ) r=448$$

Solve for $$r$$ .

$$r=\dfrac{448\times 7}{2\times 22}$$

$$r=\dfrac{784}{11}$$

$$r\approx 71.27$$

Thus, the radius of the given circle is $$\dfrac{784}{11}$$ units or $$71.27$$ (approx) units.

(a)The radius of the circle is given as $$14\;{\rm{mm}}$$ .

The area of the circle is,

$$A = \pi {r^2}$$

    $$= \dfrac{{22}}{7} \times {\left( {14} \right)^2}$$

    $$= 616\;{\rm{m}}{{\rm{m}}^2}$$

(b) The diameter of the circle is given as $$49\;{\rm{m}}$$ .

The radius of the circle becomes $$\dfrac{{49}}{2}\;{\rm{m}}$$ .

The area of the circle is,

$$A = \pi {r^2}$$

      $$= \dfrac{{22}}{7} \times {\left( {\dfrac{{49}}{2}} \right)^2}$$

      $$= 1886.5\;{{\rm{m}}^2}$$

(c) radius of the circle is given as $$5\;{\rm{cm}}$$ .

The area of the circle is,

$$A = \pi {r^2}$$

      $$= \dfrac{{22}}{7} \times {\left( 5 \right)^2}$$

      $$= 78.5\;{\rm{c}}{{\rm{m}}^2}$$

$${\textbf{Step -1: Finding the total area covered by the sprinkler.}}$$

                 $${\text{Radius of the area covered by the sprinkler  =  12m}}$$

                 $$\therefore {\text{Total area covered by sprinkler is  =  }}\pi {{\text{r}}^2} = 3.14 \times {12^2}$$

                $${\text{ = 452}}{\text{.16}}{{\text{m}}^2}$$

                $${\text{Since the sprinkler covers more area than the area of garden}}$$ $${314 \text{ m}^2}$$

                $${\text{So, the sprinkler will water the entire garden.}}$$  

$${\textbf{Hence, the sprinkler will water the entire garden.}}$$

Given that,

Diameter $$D=70\,m$$

We know that,

The area of circular horizontal surface

  $$A=\pi {{r}^{2}}$$

  $$A=3.14\times 35\times 35$$

  $$A=3847\,{{m}^{2}}$$

Hence, this is the required solution

$$\textbf{Step -1: Find the area of given parallelogram.}$$

                $$\text{Base}=7\text{ cm}$$

                $$\text{Height}=4\text{ cm}$$

                $${\because\text{Area of parallelogram}=\text{Base}\times\text{Height}}$$

                $$\therefore\text{Area of given parallelogram}=7\times4$$

                                                                          $$=28\mathrm{cm^2}$$

$$\textbf{Hence, the area of given parallelogram is }\mathbf{28cm^2.}$$

$$\textbf{Step -1: Find the area of given parallelogram.}$$

                $$\text{Base}=5\text{ cm}$$

                $$\text{Height}=3\text{ cm}$$

                $${\because\text{Area of parallelogram}=\text{Base}\times\text{Height}}$$

                $$\therefore\text{Area of given parallelogram}=5\times3$$

                                                                          $$=15\mathrm{cm^2}$$

$$\textbf{Hence, the area of given parallelogram is }\mathbf{15cm^2.}$$

$$\textbf{Step -1: Find the area of given parallelogram.}$$

                $$\text{Base}=2.5\text{ cm}$$

                $$\text{Height}=3.5\text{ cm}$$

                $${\because\text{Area of parallelogram}=\text{Base}\times\text{Height}}$$

                $$\therefore\text{Area of given parallelogram}=2.5\times3.5$$

                                                                          $$=8.75\mathrm{cm^2}$$

$$\textbf{Hence, the area of given parallelogram is }\mathbf{8.75cm^2.}$$

$$\textbf{Step -1: Find the area of given parallelogram.}$$

                $$\text{Base}=5\text{ cm}$$

                $$\text{Height}=4.8\text{ cm}$$

                $${\because\text{Area of parallelogram}=\text{Base}\times\text{Height}}$$

                $$\therefore\text{Area of given parallelogram}=5\times4.8$$

                                                                        $$=24\mathrm{cm^2}$$

$$\textbf{Hence, the area of given parallelogram is }\mathbf{24cm^2.}$$

$$\textbf{Step -1: Find the area of given parallelogram.}$$

                $$\text{Base}=2\text{ cm}$$

                $$\text{Height}=4.4\text{ cm}$$

                $${\because\text{Area of parallelogram}=\text{Base}\times\text{Height}}$$

                $$\therefore\text{Area of given parallelogram}=2\times4.4$$

                                                                          $$=8.8\mathrm{ cm^2}$$

$$\textbf{Hence, the area of given parallelogram is }\mathbf{8.8cm^2.}$$

Given that, area of a circle is $$16$$ times the area of another circle with diameter $$1.4 \ m$$ .

To find out: Circumference of the first circle.

Now, radius of the second circle $$=\frac{1.4}{2}=0.7\ m$$

We know that, area of a circle $$=\pi r^2$$

$$\therefore \ $$ Area of the second circle $$=\frac{22}{7}\times 0.7\times 0.7$$

$$\Rightarrow 1.54\ m^2$$

Then,

Area of first circle $$=16\times 1.54=24.64 \ m^2$$

$$\therefore \ \pi {{r}^{2}}=24.64$$

$${{r}^{2}}=\dfrac{24.64}{\pi }$$

$${{r}^{2}}=\dfrac{24.64}{\dfrac{22}{7}}$$

$${{r}^{2}}=\dfrac{24.64\times 7}{22}$$

$${{r}^{2}}=7.84\,$$

$$r=\sqrt{7.84}$$

$$r=2.8\ m$$

Now, Circumference of circle $$=2\pi r$$

$$=2\times \dfrac{22}{7}\times 2.8$$

$$=\dfrac{123}{7}$$

$$=17.6\ m$$

Hence, the circumference of the circle is $$17.6 \ m$$ .

We have,

Radius of circle $$=14\,cm.$$

Length of arc $$=22\,cm.$$

Using formula,

$$Angle=\dfrac{radius}{arc}$$

Now,

$$Angle=\dfrac{22}{14}$$

$$Angle=\dfrac{22}{2\times 7}$$

$$Angle=\dfrac{\pi }{2}\,\,\,\,\left( \pi =\dfrac{22}{7} \right)$$

$$Angle=\theta =\dfrac{\pi }{2}$$

So,

Area of sector $$=\dfrac{\theta }{{{360}^{o}}}\times \pi {{r}^{2}}$$

$$=\dfrac{{{90}^{o}}}{{{360}^{o}}}\times \dfrac{22}{7}\times 14\times 14$$

$$=\dfrac{1}{4}\times 22\times 2\times 14$$

$$=22\times 7$$

$$= 154\,sq.\,cm.$$

Hence, this is the answer.

Let the diameter of the circle be $$=D$$

Now, Circumference of the circle $$=\pi \times Diameter$$

According to given question,

$$\pi \times D=\pi +D$$

$$\dfrac{22}{7}D=\dfrac{22}{7}+D$$

$$\dfrac{22}{7}D=\dfrac{22+7D}{7}$$

$$22D=22+7D$$

$$22D-7D=22$$

$$15D=22$$

$$D=\dfrac{22}{15}$$

Then,

$$D=x=\dfrac{22}{15}$$

We have,

$$\left( {{x}_{1}},{{y}_{1}} \right)=\left( 2,3 \right)$$

$$\left( x,{{y}_{2}} \right)=\left( -1,0 \right)$$

$$\left( {{x}_{3}},{{y}_{3}} \right)=\left( 2,-4 \right)$$

We know that,

Area of triangle

$$=\dfrac{1}{2}\left| \left[ {{x}_{1}}\left( {{y}_{2}}-{{y}_{3}} \right)+{{x}_{2}}\left( {{y}_{3}}-{{y}_{1}} \right)+{{x}_{3}}\left( {{y}_{1}}-{{y}_{2}} \right) \right] \right|$$

$$=\dfrac{1}{2}\left| \left[ 2\left( 0+4 \right)-1\left( -4-3 \right)+2\left( 3-0 \right) \right] \right|$$

$$=\dfrac{1}{2}\left| \left[ 8+7+6 \right] \right|$$

$$=\dfrac{1}{2}\left|21 \right|$$

$$= \dfrac{21}{2}\,\text{sq}\text{.}\,\text{unit}$$

$$\textbf{Hint: Use area of circle as}$$ $$\pi {{r}^{2}}$$ and circumference as $$2\pi r$$

$$\textbf{Step-1: Finding radius of required circle.}$$

                $$\text{Diameter of given circle = 7cm, Radius = 7/2 cm}$$

                $$\text{Area of required circle is }$$

                $$\Rightarrow \dfrac{49}{4}\pi \times 16$$

                $$\Rightarrow 196\pi c{{m}^{2}}$$

$$\textbf{Step-2: Radius of required circle is -}$$

                $$Radius=\sqrt{\dfrac{Area}{\pi }}=\sqrt{\dfrac{196\pi }{\pi }}$$

                $$\Rightarrow \sqrt{196}=14cm$$

                $$Radius=\sqrt{\dfrac{Area}{\pi }}=\sqrt{\dfrac{196\pi }{\pi }}$$

                $$\Rightarrow \sqrt{196}=14cm$$

$$\textbf{Step-3: Circumference of required circle is –}$$

                 $$\therefore Circumference=2\pi r=2\times \dfrac{22}{7}\times 14$$

                 $$\Rightarrow 88cm$$

$$\textbf{Final Answer: Therefore, the circumference of cicle whose area is 16 times the area of}$$

$$\textbf{circle with diameter as 7 cm is 88 cm.}$$