Mathematics — CBSE Class 10 board question

Given: Radius of circular land, r = 35 m

(i) Length of wire needed to fence the land (Circumference):

$$C = 2\pi r = 2 \times \frac{22}{7} \times 35 = 220 \text{ m}$$

(ii) Side of the largest square inscribed in the circle:

The diagonal of the square = diameter of circle = 2 × 35 = 70 m

$$\text{Side} = \frac{\text{diagonal}}{\sqrt{2}} = \frac{70}{\sqrt{2}} = 35\sqrt{2} \text{ m}$$

(iii) Area of shaded region = Area of circle − Area of square

$$= \pi r^2 - \text{side}^2 = \frac{22}{7} \times 35 \times 35 - (35\sqrt{2})^2$$

$$= 3850 - 2450 = 1400 \text{ m}^2$$

$$\text{Cost} = 1400 \times 50 = ₹70{,}000$$

Source: Areas Related to Circles, Chapter 11

---

• (i) Fence = circumference; use $2\pi r$ with $\pi = 22/7$.
• (ii) Key insight: the diagonal of the inscribed square equals the diameter. Use diagonal = $a\sqrt{2}$, so $a = 70/\sqrt{2} = 35\sqrt{2}$.
• (iii) Shaded area = circle area − square area. Note $(35\sqrt{2})^2 = 35^2 \times 2 = 2450$. Multiply by rate ₹50 for cost. These steps must be shown for full marks.