NCERT Solutions for Class 10 Maths Chapter 12 Areas Related To Circles (Ex 12.3) Exercise 12.3

Exercise 12.3

1. Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the center of the circle. $pi =\dfrac{22}{7}$

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Ans: From the given figure, 

RQ is the diameter of the circle which implies that \[\angle RPQ = {90^ \circ }\] 

Thus, 

By applying Pythagoras theorem in \[PQR\], 

\[ \Rightarrow R{P^2} + P{Q^2} = R{Q^2}\]

\[ \Rightarrow {(7)^2} + {(24)^2} = R{Q^2}\]

\[ \Rightarrow RQ = 25\]

Thus, Radius of circle, \[OR = \frac{{RQ}}{2} = \frac{{25}}{2}\]

We know that,  RQ is the diameter of the circle, it divides the circle in two equal parts. 

So, for area of shaded region,

Area of semicircle \[ = \frac{1}{2}\pi {r^2}\]

                               \[ = \frac{1}{2}\pi {\left( {\frac{{25}}{2}} \right)^2}\]

                                \[ = \frac{{6875}}{{28}}c{m^2}\] 

 Area of \[PQR\] \[ = \frac{1}{2} \times PQ \times PR\]

                         \[ = \frac{1}{2} \times 24 \times 7\]

                          \[ = 84c{m^2}\]

Area of shaded region = Area of semi - circle RPQOR − Area of \[PQR\]

                                     \[ = \frac{{6875}}{{28}} - 84 = \frac{{4532}}{{28}}c{m^2}\]

Therefore, the area of the shaded region in the given figure is \[161.85 c{m^2}\].

2. Find the area of the shaded region in the given figure, if radii of the two concentric circles with center O are 7 cm and 14 cm respectively and \[\angle AOC = {40^ \circ }\]. $pi =\dfrac{22}{7}$

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Ans:

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Given that,

Radius of inner circle = 7 cm

Radius of outer circle = 14 cm 

Angle subtended is \[{40^ \circ }\]

Now,

Area of shaded region = Area of sector OAFC − Area of sector OBED

                                          \[ = \frac{{{{40}^ \circ }}}{{{{360}^ \circ }}} \times \pi {(14)^2} - \frac{{{{40}^ \circ }}}{{{{360}^ \circ }}} \times \pi {(7)^2}\]

                                          \[ = \frac{{616}}{9} - \frac{{159}}{9}\]

                                        \[ = \frac{{154}}{3}c{m^2}\]

Therefore, the area of the shaded region in the given figure is \[ = 51.33 c{m^2}\].

3. Find the area of the shaded region in the given figure, if ABCD is a square of side 14 cm and APD and BPC are semicircles. $pi =\dfrac{22}{7}$

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Ans:

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From the above figure it is evident that the radius of each semi-circle is 7 cm. 

For area of shaded region,

So, Area of each semi-circle \[ = \frac{1}{2}\pi {r^2}\]

                                               \[ = \frac{1}{2} \times \frac{{22}}{7} \times {(7)^2}\]

                                               \[ = 77c{m^2}\] 

Area of square ABCD \[ = {(side)^2} = {(14)^2} = 196c{m^2}\]

Area of the shaded region = Area of square ABCD − Area of semi-circle APD − Area of semi-circle BPC

                                          \[ = 196 - 77 - 77 = 42c{m^2}\]

Therefore, the area of the shaded region in the given figure is \[42c{m^2}\].

4. Find the area of the shaded region in the given figure, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as center. $pi =\dfrac{22}{7}$

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 Ans:

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Given that,

Radius of the circle is 6cm 

We know that each interior angle of an equilateral triangle is of measure \[{60^ \circ }\].

For area of shaded region,

Area of sector OCDE \[ = \frac{{{{60}^ \circ }}}{{{{360}^ \circ }}}\pi {r^2}\]

                                    \[ = \frac{1}{6} \times \frac{{22}}{7} \times {(6)^2}\]

                                    \[ = \frac{{132}}{7}c{m^2}\]

Area of \[OAB\] \[ = \frac{{\sqrt 3 }}{4}{(12)^2}\]

                        \[ = 36\sqrt 3 c{m^2}\]

Area of circle \[ = \pi {r^2}\]

                       \[ = \frac{{22}}{7} \times {(6)^2}\]

                       \[ = \frac{{792}}{7}c{m^2}\]

Area of shaded region = Area of ΔOAB + Area of circle − Area of sector OCDE

                                    \[ = 36\sqrt 3  + \frac{{792}}{7} - \frac{{132}}{7}\]

                                    \[ = \left( {36\sqrt 3  + \frac{{660}}{7}} \right)c{m^2}\]

Hence, the area of the shaded region in the given figure \[\left( {36\sqrt 3  + \frac{{660}}{7}} \right)c{m^2}\].

5. From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in the given figure. Find the area of the remaining portion of the square. $pi =\dfrac{22}{7}$

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Ans:

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It is evident from the above figure that each quadrant is a sector of \[{90^ \circ }\] in a circle of 1 cm radius.

For area of shaded region,

Area of each quadrant \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}}\pi {r^2}\]

                                     \[ = \frac{1}{4} \times \frac{{22}}{7} \times {(1)^2}\]

                                     \[ = \frac{{22}}{{28}}c{m^2}\] 

Area of square \[ = {(side)^2}\]

                         \[ = {(4)^2}\]

                         \[ = 16c{m^2}\]

Area of circle \[ = \pi {r^2}\]

                       \[ = \pi {(1)^2}\] \[ = \frac{{22}}{7}c{m^2}\]

Area of the shaded region = Area of square − Area of circle – (4 × Area of quadrant)

                                          \[ = 16 - \frac{{22}}{7} - \left( {4 \times \frac{{22}}{{28}}} \right)\]

                                           \[ = 16 - \frac{{44}}{7}\]

                                            \[ = \frac{{68}}{7}c{m^2}\]  

Therefore, the area of the remaining portion of the square is \[\frac{{68}}{7}c{m^2}\].

6. In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the given figure. Find the area of the design (Shaded region). $pi =\dfrac{22}{7}$

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Ans:

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Given that,

Radius of circle = \[r\] = 32 cm 

AD is the median of \[ABC\] 

\[OA = \frac{2}{3}AD\]

\[AD = 48cm\] 

In \[ABD\],

Using Pythagoras Theorem,

\[A{B^2} = A{D^2} + B{D^2}\]

\[A{B^2} = {(48)^2} + {\left( {\frac{{BC}}{2}} \right)^2}\]

\[A{B^2} = {(48)^2} + {\left( {\frac{{AB}}{2}} \right)^2}\]

\[ \Rightarrow \frac{{3A{B^2}}}{4} = {(48)^2}\]

\[ \Rightarrow AB = \frac{{48 \times 2}}{{\sqrt 3 }} = \frac{{96}}{{\sqrt 3 }}\]

          \[ = 32\sqrt 3 \] cm

For area of design,

Area of equilateral triangle \[ = \frac{{\sqrt 3 }}{4} \times {(32)^2} \times 3\]

                                            \[ = 96 \times 8 \times \sqrt 3 \]

                                            \[ = 768\sqrt 3 c{m^2}\] 

Area of circle \[ = \pi {r^2}\]

                       \[ = \frac{{22}}{7} \times {(32)^2}\]

                       \[ = \frac{{22528}}{7}c{m^2}\] 

Area of design = Area of circle − Area of \[ABC\]

                         \[ = \left( {\frac{{22528}}{7} - 768\sqrt 3 } \right)c{m^2}\]

Hence, the area of the design (Shaded region) is \[\left( {\frac{{22528}}{7} - 768\sqrt 3 } \right)c{m^2}\].

7. In the given figure, ABCD is a square of side 14 cm. With centers A, B, C and D, four circles are drawn such that each circle touches externally two of the remaining three circles. Find the area of the shaded region. $pi =\dfrac{22}{7}$

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Ans:

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It is evident that,

Area of each of the 4 sectors is equal to each other 

Sector of \[{90^ \circ }\] in a circle of 7 cm radius.

For the area of shaded region,

Area of each sector \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}}\pi {(7)^2}\]

                                \[ = \frac{1}{4} \times \frac{{22}}{7} \times {\left( 7 \right)^2}\]

                                \[ = \frac{{77}}{2}c{m^2}\]

Area of square ABCD \[ = {(side)^2}\]

                                     \[ = {(14)^2} = 196c{m^2}\] 

Area of shaded portion = Area of square ABCD – (4 × Area of each sector)

                                     \[ = 196 - \left( {4 \times \frac{{77}}{2}} \right)\]

                                     \[ = 42c{m^2}\] 

Therefore, the area of shaded portion is \[42c{m^2}\].

8. The given figure depicts a racing track whose left and right ends are semicircular.

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The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, $pi =\dfrac{22}{7}$

find 

Ans:

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(i). The distance around the track along its inner edge

Ans: For inner edge,

Radius = \[r\]= 30cm

Distance around the track along its inner edge = AB + arc BEC + CD + arc DFA

                  \[ = 106 + \left( {\frac{1}{2} \times 2\pi r} \right) + 106 + \left( {\frac{1}{2} \times 2\pi r} \right)\]

                  \[ = 212 + \left( {\frac{1}{2} \times 2 \times \frac{{22}}{7} \times 30} \right) + \left( {\frac{1}{2} \times 2 \times \frac{{22}}{7} \times 30} \right)\]

                  \[ = 212 + \left( {2 \times \frac{{22}}{7} \times 30} \right)\]

                  \[ = \frac{{2804}}{7}m\]

                  Hence, distance around the track along its inner edge is \[\frac{{2804}}{7}m\].

(ii). The area of the track

Ans: Radius of inner edge = 30cm

   Radius of outer edge = 40cm

Area of the track = (Area of GHIJ − Area of ABCD) + (Area of semi-circle HKI – Area of semi- circle BEC) + (Area of semi-circle GLJ − Area of semicircle AFD)

\[ = \left( {106 \times 80} \right) - \left( {106 \times 60} \right) + \left( {\frac{1}{2} \times \frac{{22}}{7} \times {{(40)}^2}} \right) - \left( {\frac{1}{2} \times \frac{{22}}{7} \times {{(30)}^2}} \right) + \left( {\frac{1}{2} \times \frac{{22}}{7} \times {{(40)}^2}} \right) - \left( {\frac{1}{2} \times \frac{{22}}{7} \times {{(30)}^2}} \right)\]

\[ = 106\left( {80 - 60} \right) + \left( {\frac{{22}}{7} \times {{\left( {40} \right)}^2}} \right) - \left( {\frac{{22}}{7} \times {{\left( {30} \right)}^2}} \right)\]

\[ = 2120 + \frac{{22}}{7}\left[ {{{\left( {40} \right)}^2} - {{\left( {30} \right)}^2}} \right]\]

\[ = 2120 + 2200\]

 \[ = 4320{m^2}\]

Therefore, the area of the track is \[4320{m^2}\].

9. In the given figure, AB and CD are two diameters of a circle (with center O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

$pi =\dfrac{22}{7}$

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Ans:

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Given that,

Radius of larger circle \[ = {r_1}\] = 7 cm 

Radius of smaller circle  \[ = {r_2}\] = \[\frac{7}{2}\] cm 

For area of shaded region,

Area of smaller circle \[ = \pi {r_2}^2\]

                                    \[ = \frac{{22}}{7} \times {\left( {\frac{7}{2}} \right)^2}\]

                                    \[ = \frac{{77}}{2}c{m^2}\]

Area of semi-circle AECFB of larger circle \[ = \frac{1}{2}\pi {r_2}^2\]

                                                                      \[ = \frac{1}{2} \times \frac{{22}}{7} \times {\left( 7 \right)^2}\]

                                                                      \[ = \frac{{77}}{2}c{m^2}\]

Area of \[ABC\]\[ = \frac{1}{2} \times AB \times OC\]

                       \[ = \frac{1}{2} \times 14 \times 7\]

                       \[ = 49c{m^2}\]

Area of the shaded region = Area of smaller circle + Area of semi-circle AECFB − Area of \[ABC\]

                                           \[ = \frac{{77}}{2} + 77 - 49\]

                                           \[ = 28 + 38.5 = 66.5c{m^2}\]

Therefore, the area of shaded region is \[66.5c{m^2}\].

10. The area of an equilateral triangle ABC is \[17320.5c{m^2}\]. With each vertex of the triangle as center, a circle is drawn with radius equal to half the length of the side of the triangle (See the given figure). Find the area of shaded region.  [\mathbf{Use}\text{ }\pi =3.14 \text{ }\mathbf{and}\text{ }\sqrt{3}=1.73205]

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Ans:

Let the side of the equilateral triangle be a. 

Given that,

Area of equilateral triangle \[ = 17320.5c{m^2}\]

\[ \Rightarrow \frac{{\sqrt 3 }}{4}{(a)^2} = 17320.5\]

\[ \Rightarrow \frac{{1.73205}}{4}{a^2} = 17320.5\]

\[ \Rightarrow {a^2} = 4 \times 10000\]

\[ \Rightarrow a = 200cm\]

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It is evident from the figure that, each sector is of measure \[{60^ \circ }\] 

Area of sector ADEF \[ = \frac{{{{60}^ \circ }}}{{{{360}^ \circ }}} \times \pi  \times {r^2}\]

                                   \[ = \frac{1}{6} \times \pi  \times {\left( {100} \right)^2}\]

                                   \[ = \frac{{3.14 \times 10000}}{6}\]

                                    \[ = \frac{{15700}}{3}c{m^2}\]

Area of shaded region = Area of equilateral triangle – (3 × Area of each sector)

                                    \[ = 17320.5 - 3 \times \frac{{15700}}{3}\]

                                    \[ = 17320.5 - 15700\]

                                    \[ = 1620.5c{m^2}\]

Therefore, area of given shaded region is \[1620.5c{m^2}\].

11. On a square handkerchief, nine circular designs each of radius 7 cm are made (see the given figure). Find the area of the remaining portion of the handkerchief. $pi =\dfrac{22}{7}$

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Ans:

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It is evident from the above figure, that the side of the square is 42 cm. 

So, for area of the remaining portion of handkerchief,

Area of square \[ = {(side)^2} = {(42)^2}\]

                         \[ = 1764c{m^2}\]

Area of each circle \[ = \pi {r^2}\]\[ = \frac{{22}}{7} \times {(7)^2}\]

                               \[ = 154c{m^2}\] 

Area of 9 circles \[ = 9 \times 154\]

                            \[ = 1386c{m^2}\] 

Area of the remaining portion of the handkerchief \[ = 1764 - 1386\]

                                                                                 \[ = 378c{m^2}\]

Therefore, the area of the remaining portion of the handkerchief is  \[ = 378c{m^2}\].

12. In the given figure, OACB is a quadrant of circle with center O and radius 3.5 cm. If OD = 2 cm, find the area of the $pi =\dfrac{22}{7}$

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Ans:

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Given that radius is 3.5cm.

1. Quadrant OACB

Ans:

Since OACB is a quadrant, the angle at O is \[{90^ \circ }\]. 

Area of quadrant OACB \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}} \times \pi {r^2}\]

                                        \[ = \frac{1}{4} \times \frac{{22}}{7} \times {\left( {3.5} \right)^2}\]

                                        \[ = \frac{1}{4} \times \frac{{22}}{7} \times {\left( {\frac{7}{2}} \right)^2}\]

                                        \[ = \frac{{77}}{8}c{m^2}\]

Therefore, the area of quadrant OACB is \[\frac{{77}}{8}c{m^2}\].

2. Shaded region

Ans:

For the area of the shaded region,

Area of \[OBD\] \[ = \frac{1}{2} \times OB \times OD\]

                        \[ = \frac{1}{2} \times 3.5 \times 2\]

                        \[ = \frac{7}{2}c{m^2}\]   

Area of the shaded region = Area of quadrant OACB − Area of \[OBD\] 

                                          \[ = \frac{{77}}{8} - \frac{7}{2}\]

                                          \[ = \frac{{49}}{8}c{m^2}\].

Hence, the area of the shaded region in the given figure is \[\frac{{49}}{8}c{m^2}\].

13. In the given figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. [\mathbf{Use}\text{ }\pi =3.14]

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Ans:

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For radius

In \[OAB\],

Using Pythagoras Theorem,

\[O{B^2} = O{A^2} + O{B^2}\]

       \[ = {(20)^2} + {(20)^2}\]

\[ \Rightarrow OB = 20\sqrt 2 \]

Radius of circle \[ = r = \]\[20\sqrt 2 \] cm 

For the area of shaded region,

Area of quadrant OPBQ \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}} \times 3.14 \times {\left( {20\sqrt 2 } \right)^2}\]

                                        \[ = \frac{1}{4} \times 3.14 \times 800\]

                                        \[ = 628c{m^2}\]

Area of square OABC \[ = {\left( {side} \right)^2}\]

                                     \[ = {(20)^2}\]

                                     \[ = 400c{m^2}\] 

Area of shaded region = Area of quadrant OPBQ − Area of square OABC

                                    \[ = \left( {628 - 400} \right)c{m^2}\]

                                    \[ = 228c{m^2}\]

Therefore, the area of the shaded region is \[228c{m^2}\].

14. AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and center O (see the given figure). If \[\angle AOB = {30^ \circ }\], find the area of the shaded region. $pi =\dfrac{22}{7}$

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Ans:

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Given that,

Radius for sector OAEB = 21cm

Radius for sector OCFD = 7cm

Angle subtended is \[{30^ \circ }\]

Area of the shaded region = Area of sector OAEB − Area of sector OCFD 

                                          \[ = \left[ {\frac{{{{30}^ \circ }}}{{{{360}^ \circ }}} \times \pi  \times {{\left( {21} \right)}^2}} \right] - \left[ {\frac{{{{30}^ \circ }}}{{{{360}^ \circ }}} \times \pi  \times {{\left( 7 \right)}^2}} \right]\]

                                          \[ = \frac{1}{{12}}\pi \left[ {\left( {21 - 7} \right)\left( {21 + 7} \right)} \right]\]

                                          \[ = \frac{{22 \times 14 \times 28}}{{12 \times 7}} = \frac{{22 \times 14 \times 28}}{{12 \times 7}}\]

                                          \[ = \frac{{308}}{3}c{m^2}\]

Therefore, the area of shaded region is \[\frac{{308}}{3}c{m^2}\].

15. In the given figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region. $pi =\dfrac{22}{7}$

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Ans:

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Given that,

Radius of circle = \[r = \] 14cm

As ABC is a quadrant of the circle, \[\angle BAC\] will be of measure \[{90^ \circ }\]. 

In \[ABC\], 

Using Pythagoras Theorem,

\[ \Rightarrow B{C^2} = A{C^2} + A{B^2}\]

            \[ = {(14)^2} + {(14)^2}\]

\[ \Rightarrow BC = 14\sqrt 2 \] 

Radius of semi-circle drawn on BC = \[{r_1}\] \[ = \frac{{14\sqrt 2 }}{2} = 7\sqrt 2 cm\]

Now, for area of the shaded region,

Area of \[ABC\] \[ = \frac{1}{2} \times AB \times AC\]

                        \[ = \frac{1}{2} \times {(14)^2}\]

                        \[ = 98c{m^2}\]

Area of sector ABDC \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}} \times \pi {r^2}\]

                                    \[ = \frac{1}{4} \times \frac{{22}}{7} \times {(14)^2}\]

                                    \[ = 154c{m^2}\]

Area of semi-circle drawn on BC \[ = \frac{1}{2} \times \pi  \times {r_1}^2\]

                                                      \[ = \frac{1}{2} \times \frac{{22}}{7} \times {\left( {7\sqrt 2 } \right)^2}\]

                                                      \[ = 154c{m^2}\]

Area of shaded region = Area of semi-circle on BC – (Area of sector ABDC – Area of ∆ABC)

                                     \[ = 154 - \left( {154 - 98} \right)\]

                                     \[ = 98c{m^2}\]

Hence, the area of the shaded region of the given figure is \[98c{m^2}\].

16. Calculate the area of the designed region in the given figure common between the two quadrants of circles of radius 8 cm each. $pi =\dfrac{22}{7}$

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Ans:

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Given that ,

Radius of each circle is 8cm

The designed area is the common region between two sectors BAEC and DAFC.

For the area of the designed region,

Area of sector \[ = \frac{{{{90}^ \circ }}}{{{{360}^ \circ }}} \times \frac{{22}}{7} \times {\left( 8 \right)^2}\]

                        \[ = \frac{1}{4} \times \frac{{22}}{7} \times 64\]

                        \[ = \frac{{22 \times 16}}{7}\]

                        \[ = \frac{{352}}{7}c{m^2}\] 

Area of \[BAC\] \[ = \frac{1}{2} \times BA \times BC\]

                        \[ = \frac{1}{2} \times {(8)^2}\]

                        \[ = 32c{m^2}\] 

Area of the designed portion = 2 × (Area of segment AEC) = 2 × (Area of sector BAEC − Area of \[BAC\])

                                               \[ = 2 \times \left( {\frac{{352}}{7} - 32} \right)\]

                                               \[ = \frac{{2 \times 128}}{7}\] 

                                               \[ = \frac{{256}}{7}c{m^2}\]

Hence, the area of the designed region is \[\frac{{256}}{7}c{m^2}\]. 

NCERT Solutions for Class 10 Maths Chapter 12 Areas Related To Circles Exercise 12.3

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