Question

the circular path is outside the circular garden. if its external circumference is 508m and internal circumference is 420 m. what is the area of the path​

Answer 1

Answer:

$\bigstar{\bold{Area\:of\:the\:path=6496.16\:m^{2} }}$

Explanation:

$\Large{\underline{\bf{Given:}}}$

• External circumference of the path = 508 m
• Internal circumference of the path = 420 m

$\Large{\underline{\bf{To\:Find:}}}$

• Area of the path

$\Large{\underline{\bf{Solution:}}}$

➻ Here we have to find area of the circular path.

➻ First let us find the radius of the outer and inner circles.

➻ Let the radius of the outer circle be R and inner circle be r.

                    $\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\qbezier(2.3,0)(2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,2.121)(0,2.3)\qbezier(-2.3,0)(-2.121,-2.121)(0,-2.3)\qbezier(2.3,0)(2.121,-2.121)(-0,-2.3)\qbezier(1.2,0)(1.121,1.121)(0,1.2)\qbezier(1.2,0)(1.121,-1.121)(0,-1.2)\qbezier(0,-1.2)(-1.121,-1.121)(-1.2,0)\qbezier(-1.2,0)(-1.121,1.121)(0,1.2)\put(-0,0){\vector(-1,0){2.3}}\put(0,0){\vector(0,1){1.2}}\put(-1.8,0.2){ \bf R }\put(0.2,0.3){ \bf\large r }\end{picture}$

➻ We are given the circumference/perimeter of the two circles.

➻ The circumference of a circle is given by,

    Circumference of a circle = 2 π r

➻ Hence,

    Circumference of outer circle = 508 m

     2 π R = 508 m

     R = 508 × 7/44

     R = 80.82 m

➻ Therefore radius of the outer circle is 80.82 m

➻ Now finding the area of the outer circle,

➻ Area of a circle is given by,

    Area of a circle = π r²

➻ Hence,

    Area of outer circle = 22/7 × 80.82 × 80.82

   Area of outer circle = 20528.74 m²

➻ Now finding the radius of the inner circle,

    Circumference of inner circle = 420 m

    2 π r = 420

    r = 66.82 m

➻ Now finding the area of inner circle,

    Area of inner circle = 22/7 × 66.82 × 66.82

    Area of inner circle = 14032.58 m²

➻ Now area of the path is given by,

   Area of the path = Area of outer circle - Area of inner circle

➻ Substitute the data,

    Area of the path = 20528.74 - 14032.58

    Area of the path = 6496.16

➻ Hence area of the path is 6496.16 m.²

    $\boxed{\bold{Area\:of\:the\:path=6496.16\:m^{2} }}$

   

Answer 2

Given:-

♱External Circumference = 508m

♱Internal Circumference = 420m

Find:-

⚚Area of the path

Solution:-

For outer circle

we, know that

$\huge{\underline{\boxed{\sf Circumference \: of \: circle = 2 \pi r}}}$

where,

• Circumference of outer circle = 508m
• π = 22/7

So,

$\dashrightarrow \sf Circumference_{outer \: circle}= 2 \pi r$

$\dashrightarrow \sf 508= 2 \times \dfrac{22}{7} r$

$\dashrightarrow \sf 508=\dfrac{44}{7} r$

$\dashrightarrow \sf 508 \times \dfrac{7}{44} =r$

$\dashrightarrow \sf \dfrac{3556}{44} =r$

$\dashrightarrow \sf 80.818(approx.) m=r$

$\dashrightarrow \sf 80.8m=r$

$\therefore \sf radius_{outer \: circle} = 80.8m$

Now, using

$\huge{\underline{\boxed{\sf Area \: of \: circle =\pi r^2}}}$

where,

• Radius of outer circle = 80.8m

So,

$\dashrightarrow\sf Area_{outer \: circle}=\pi r^2$

$\dashrightarrow\sf Area_{outer \: circle}= \dfrac{22}{7} \times (80.8)^2$

$\dashrightarrow\sf Area_{outer \: circle}= \dfrac{22}{7} \times 6528.64$

$\dashrightarrow\sf Area_{outer \: circle}= \dfrac{143630.08}{7}$

$\dashrightarrow\sf Area_{outer \: circle}=20518.582 {m}^{2} (approx.)$

$\dashrightarrow\sf Area_{outer \: circle}=20518.6 {m}^{2}$

$\small{\therefore\sf Area_{outer \: circle}=20518.6 {m}^{2}}$

Now, for inner circle

$\huge{\underline{\boxed{\sf Circumference \: of \: circle = 2 \pi r}}}$

where,

• Circumference of inner circle = 420m

So,

$\dashrightarrow \sf Circumference_{inner \: circle}= 2 \pi r$

$\dashrightarrow \sf 420= 2 \times \dfrac{22}{7} r$

$\dashrightarrow \sf 420= \dfrac{44}{7} r$

$\dashrightarrow \sf 420 \times \dfrac{7}{44} = r$

$\dashrightarrow \sf \dfrac{2940}{44} = r$

$\dashrightarrow \sf 66.818m(approx.)= r$

$\dashrightarrow \sf 66.82m= r$

$\therefore\sf radius_{inner \: circle}= 66.82m$

Again, using

$\huge{\underline{\boxed{\sf Area \: of \: circle =\pi r^2}}}$

where,

• Radius of Inner Circle = 66.82m

So,

$\dashrightarrow\sf Area_{inner\: circle}=\pi r^2$

$\dashrightarrow\sf Area_{inner\: circle}= \dfrac{22}{7} \times (66.82)^2$

$\dashrightarrow\sf Area_{inner\: circle}= \dfrac{22}{7} \times 4464.9124$

$\dashrightarrow\sf Area_{inner\: circle}= \dfrac{98228.0728}{7}$

$\dashrightarrow\sf Area_{inner\: circle}= 14032.581m^2(approx.)$

$\dashrightarrow\sf Area_{inner\: circle}= 14032.58m^2$

$\therefore\sf Area_{inner\: circle}= 14032.58m^2$

Now,

➙Area of path = Area of outer circle - Area of Inner Circle

▶20518.6m² - 14032.58m²

▶6486.02m²

Hence, Area of the path is 6486.02m²