Ungrazed area with four tethered horses at a square’s corners: Four horses are tethered at the four corners of a square plot of side 63 m, each with a rope just long enough that adjacent horses cannot reach one another. Find the ungrazed area.

Introduction / Context:
This classic geometry problem involves subtracting grazed circular sectors from the area of a square. “Just cannot reach one another” implies each rope length equals half the side, so the grazed regions touch but do not overlap across the square’s interior.

Given Data / Assumptions:

• Square side a = 63 m
• Each rope length r = a/2 = 31.5 m
• Each horse grazes a quarter-circle sector inside the square

Concept / Approach:
Total grazed area = 4 * (1/4 * π * r^2) = π * r^2. Ungrazed area = area(square) − grazed area.

Step-by-Step Solution:
Area(square) = a^2 = 63^2 = 3969 m2r = 31.5 m → r^2 = 992.25Grazed = π * 992.25 = (22/7) * 992.25 = 3118.5 m2Ungrazed = 3969 − 3118.5 = 850.5 m2

Verification / Alternative check:
Using π ≈ 3.1416 yields ~851.7 m2; standard test convention uses π = 22/7 giving the exact listed value 850.5 m2.

Why Other Options Are Wrong:
Other values do not match the precise subtraction using quarter-circle sectors with r = 31.5 m and a = 63 m.

Common Pitfalls:
Assuming r = 63 m instead of 31.5 m; forgetting there are four quarter-circles (which sum to one full circle).

Final Answer:
850.5 m2