Question

A circular field has a perimeter of 650 m. A square plot having its vertices on the circumference of the field is marked in the field. Calculate the area of the square plot.

Answer 1

Answer:

The Area of the square plot is 21387 m².

Step-by-step explanation:

Given :  

Perimeter of a circular field = 650 m

Circumference (Perimeter) of circle = 2πr

650 = 2 × π × r

2πr = 650

r = 650/2π

r = 325/π m

Diagonal of the square plot = Diameter of the circle.

Diagonal of the square plot = 2 × r  

= 2 × 325/π m

Diagonal of the square plot = 650/π m

Area of the square plot = 1/2 × (Diagonal of the square)²

= 1/2 × (650/π)²

= (½ × 650 × 650)/(22/7)²

= (325 × 650 × 49)/ 484

= 10351250/484

= 21386.88 ~ 21387 (approximately)

Area of the square plot = 21387 m²

Hence, the Area of the square plot is 21387 m².

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Answer 2

$\sf{\underline{Note:}}$ The square is inscribed in the circle.

$\sf{\underline{Here:}}$

We have to find the circumference of circle and then the area of the square plot.

$\sf{\underline{Formula\:used:}}$

$\boxed{\sf{Circumference = 2 \pi r}}$

$\sf{\underline{Now:}}$

$\sf{\underline{Circumference\:of\:Circle:}}$

$\implies$ $\sf{Circumference = 2 \pi r}$

$\implies$ $\sf{650 = \frac{2 \times 22}{7 \times r}}$

$\implies$ $\sf{r = \frac{(650 \times 7)}{44}}$

$\implies$ $\sf{r = \frac{4550}{44}}$

$\implies$ $\sf{r = 103.409}$

$\sf{\underline{As:}}$

The diagonal of the square plot is the diameter of the circle.

$\sf{\underline{Therefore:}}$ $\boxed{\sf{r \times 2 = d}}$

$\sf{\underline{Diameter:}}$

$\implies$ $\sf{103.409 \times 2}$

$\implies$ $\sf{206.818}$

$\sf{\underline{Note:}}$

$\bullet$ Diameter of circle = 206.818 m

$\bullet$ 206.818 m = Diagonal of square plot

$\sf{\underline{Now:}}$

$\sf{Area \: of \: the \: square \: plot = \frac{1}{2} \times {d}^{2}}$

$\implies$ $\sf{ \frac{1}{2} \times {(206.818)}^{2}}$

$\implies$ $\sf{ \frac{1}{2} \times 42773.68}$

$\implies$ $\sf{ 21386.84}$

$\sf{\underline{Therefore:}}$

Area of the square plot is 21387 $\sf{m^{2}}$