Rectangular Field Dimensions and Area Calculation
The problem provides the dimensions of a rectangular field: length ($l$) = 169 m and width ($w$) = 154 m. The first step is to calculate the area of this rectangular field.
The formula for the area of a rectangle is:
$$ A_{\text{rectangle}} = \text{length} \times \text{width} $$Substituting the given values:
$$ A_{\text{rectangle}} = 169 \, \text{m} \times 154 \, \text{m} $$ $$ A_{\text{rectangle}} = 26026 \, \text{m}^2 $$So, the area of the rectangular field is 26026 square meters.
Equating Areas and Finding Circular Field Radius
The problem states that the area of the rectangular field is equal to the area of a circular field. We know the area of the rectangle is $26026 \, \text{m}^2$. The formula for the area of a circle is:
$$ A_{\text{circle}} = \pi r^2 $$where $r$ is the radius of the circular field and $\pi = \frac{22}{7}$.
Since the areas are equal:
$$ A_{\text{circle}} = A_{\text{rectangle}} $$ $$ \pi r^2 = 26026 $$Now, we substitute the value of $\pi$ and solve for $r$:
$$ \frac{22}{7} r^2 = 26026 $$To find $r^2$, we rearrange the equation:
$$ r^2 = 26026 \times \frac{7}{22} $$Performing the division:
$$ r^2 = 1183 \times 7 $$ $$ r^2 = 8281 $$Now, we find the radius $r$ by taking the square root of $r^2$:
$$ r = \sqrt{8281} \, \text{m} $$ $$ r = 91 \, \text{m} $$The radius of the circular field is 91 meters.
Calculating the Circumference of the Circular Field
The final step is to calculate the circumference of the circular field using its radius. The formula for the circumference of a circle is:
$$ C_{\text{circle}} = 2 \pi r $$Substitute the values of $\pi$ and $r$:
$$ C_{\text{circle}} = 2 \times \frac{22}{7} \times 91 \, \text{m} $$Simplify the calculation:
$$ C_{\text{circle}} = 2 \times 22 \times \frac{91}{7} \, \text{m} $$ $$ C_{\text{circle}} = 44 \times 13 \, \text{m} $$ $$ C_{\text{circle}} = 572 \, \text{m} $$Therefore, the circumference of the circular field is 572 meters.
Important Questions from 2-D Mensuration
- Find the area of a sector with a central angle of 150° in a circle with a radius of 12 cm.
- The area of a square is $2304$ cm$^2$. Its perimeter is equal to the perimeter of a regular hexagon. What is the area (in cm$^2$) of the hexagon?
Calculate the area of the triangle whose sides are 8 cm, 9 cm and 13 cm. (Rounded up to two decimal places)
- The length of a rectangular pitch is 30 m more than its breadth. Its area is $18,271$ m$^{2}$. Its breadth (in m) is:
- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
(Take $\pi = \frac{22}{7}$)
