Question

Find the area of region between two concentric circular paths are 210m and 490m

Answer 1

Given,

The perimeter of the inner circle$\[ = 210\]$

The perimeter of the outer circle$\[ = 490\]$

Solution,

Calculate the radius of the inner circle(r).

$\[\begin{array}{l} \Rightarrow 2 \times \frac{{22}}{7} \times r = 210\\ \Rightarrow r = \frac{{210 \times 7}}{{2 \times 22}}\\ \Rightarrow r \approx 33\end{array}\]$

Calculate the radius of the outer circle(r).

$\[\begin{array}{l} \Rightarrow 2 \times \frac{{22}}{7} \times R = 490\\ \Rightarrow R = \frac{{490 \times 7}}{{2 \times 22}}\\ \Rightarrow R \approx 78\end{array}\]$

Therefore, the area of the region between the concentric circular path

$\[\begin{array}{l} \Rightarrow \frac{{22}}{7} \times 78 \times 78 - \frac{{22}}{7} \times 33 \times 33\\ \Rightarrow \frac{{22}}{7} \times 6084 - \frac{{22}}{7} \times 1089\\ \Rightarrow \frac{{22}}{7}\left( {6084 - 1089} \right)\end{array}\]$

Solve further,

$\[\begin{array}{l} \Rightarrow \frac{{22}}{7} \times 4995\\ \Rightarrow 15698.57\end{array}\]$

Hence, the area of the region between the concentric circular path is$\[15698.57\]$ square meters.

Answer 2

Given:

Radius of outer circle, R = 490m

Radius of inner circle, r = 210m

To find:

The area of region between two concentric circular paths are 210m and 490m.

Formula to be used:

Area of required region = π (R² - r²)

Area of required region = π (R + r) (R - r )

Solution:

Step 1 of 2:

If the values of two concentric circular paths are given, then we can find the area of required region using the following formula,

π ( R + r ) ( R - r )

Step 2 of 2:

On substituting the given values in the formula we get,

Area of required region = π ( R + r ) ( R - r )

Area of required region = $\frac{22}{7}$ × $( 490 + 210 )$ × $( 490 - 210)$

Area of required region = $\frac{22}{7}$ × $(700)$ × $(280)$

Area of required region = $\frac{22}{7}$ × $196000$

On simplification, we get

Area of required region = 616000 m²

Final answer:

The area of region between two concentric circular paths are 210m and 490m is 616000 m².