Shortcut Trick
Area of rectangular field = 30 × 20 = 600 m2
Each corner of a rectangle is 90°, so 4 corners = 360° = 1 complete circle.
Grazed Area = Area of circle with radius 7 m = πr2
Grazed Area = (22/7) × 7 × 7 = 154 m2
Ungrazed Area = Total Area − Grazed Area = 600 − 154 = 446 m2
∴ The correct answer is 446 m2.
Alternate Method
Given:
Length of rectangular field (L) = 30 m
Width of rectangular field (W) = 20 m
Length of rope (radius, r) = 7 m
Formula Used:
Area of Rectangle = Length × Width
Area of a Sector = (θ/360) × πr2
Calculations:
⇒ Total area of the field = 30 × 20 = 600 m2
⇒ In a rectangle, the interior angle at each corner is 90°.
⇒ Area grazed by one cow = (90/360) × π × (7)2
⇒ Area grazed by one cow = (1/4) × (22/7) × 49 = 38.5 m2
⇒ Total grazed area by 4 cows = 4 × 38.5 = 154 m2
⇒ Area that cows cannot graze = Total area − Total grazed area
⇒ Area that cows cannot graze = 600 − 154 = 446 m2
∴ The correct answer is 446 m2.
Additional Information
Area of Circle and Sectors
The area of a circle is πr2. Since the sum of the angles at the corners of a rectangle is 360°, the four quadrants at the corners combine to form exactly one full circle.
General Polygon Case
If cows are tied at the corners of any quadrilateral (not just a rectangle), the sum of the interior angles is always 360°, so the total grazed area will still be πr2 if the ropes are of equal length.
Perimeter of Grazed Region
The total boundary length of the grazed area inside the field is equivalent to the circumference of one circle, which is 2πr.