A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below. The radius of circular piece of land is 35 m. Based on given information, answer the following questions: (i) Find the length of wire needed to fence the entire land. (ii) Find the length of each side of the square land on which house will be constructed. (iii) (a) The farmer wishes to grow grass on the shaded region around the house. Find the cost of growing the grass at the rate of ₹ 50 per square metre. (iii) (b) Find the ratio of area of land on which house is built to remaining area of circular piece of land.

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2025: question 36

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedAreas Related to Circles

A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below.

The radius of circular piece of land is 35 m.

Based on given information, answer the following questions:

(i)
Find the length of wire needed to fence the entire land.
1 mark
(ii)
Find the length of each side of the square land on which house will be constructed.
1 mark
(iii)
(a) The farmer wishes to grow grass on the shaded region around the house. Find the cost of growing the grass at the rate of ₹ 50 per square metre.
2 marks
OR
(iii)
(b) Find the ratio of area of land on which house is built to remaining area of circular piece of land.
2 marks

Figure in the paper: Left: a circle showing a house with two gabled sections and a tree below, drawn within the circular land. Right: a circle with a largest inscribed square (house floor plan with rooms), with the four circular segments outside the square shaded grey. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Fencing the land means going once round its boundary.

Hint 2

For the largest square in a circle, think about what the diagonal of the square is equal to.

Hint 3

Area of a segment = area of sector − area of triangle. Here each side of the square subtends $9 0^{\circ}$ at the centre.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) The length of wire for fencing equals the circumference: $2 π r = 2 \times \frac{22}{7} \times 35$
½ mark
(i) $= 220$ m
½ mark
(ii) The largest square inside a circle has its diagonal equal to the diameter. So diagonal $= 2 \times 35 = 70$ m.
½ mark
(ii) Side $= \frac{diagonal}{2} = \frac{70}{2} = 352$ m ( $\approx 49.49$ m)
½ mark
(iii)(a) The shaded region consists of two segments, each cut off by a side of the square. Each side subtends $9 0^{\circ}$ at the centre. Area $= 2 [\frac{90}{360} \times \frac{22}{7} \times 35 \times 35 - \frac{1}{2} \times 35 \times 35]$
1 mark
(iii)(a) $= 2 [962.5 - 612.5] = 2 \times 350 = 700 m^{2}$
½ mark
(iii)(a) Cost $= 700 \times ₹50 = ₹35000$
½ mark
(iii)(b) Area of square $= (352)^{2} = 2450 m^{2}$ . Area of circle $= \frac{22}{7} \times 35 \times 35 = 3850 m^{2}$ . Remaining area $= 3850 - 2450 = 1400 m^{2}$ .
1½ marks
(iii)(b) Required ratio $= \frac{2450}{1400} = \frac{7}{4}$ , i.e. $7 : 4$
½ mark

Answer: (i) 220 m; (ii) $352$ m; (iii)(a) ₹35000; (iii)(b) 7 : 4

Where the marks are

(i) Uses circumference $2 π r$ (½ mark) and gets 220 m (½ mark)
(ii) Diagonal of square = diameter = 70 m (½ mark); side $= 352$ m (½ mark)
(iii)(a) Correct expression for the area of two segments (1 mark); 700 m² (½ mark); cost ₹35000 (½ mark)
(iii)(b) Correct areas of square and remaining region, 2450 and 1400 (1½ marks); ratio 7 : 4 (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Areas Related to Circles, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/5/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 36

Hints

Step-by-step solution

Revise the chapter

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