For the inauguration of 'Earth day' week in a school, badges were given to volunteers. Organisers purchased these badges from an NGO, who made these badges in the form of a circle inscribed in a square of side 8 cm. O is the centre of the circle and ∠ AOB = 90°. Based on the above information, answer the following questions: (i) What is the area of square ABCD? (ii) What is the length of diagonal AC of square ABCD? (iii) Find the area of sector OPRQO. (iii) Find the area of remaining part of square ABCD when area of circle is excluded.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/6/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2023: question 36

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, 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

For the inauguration of 'Earth day' week in a school, badges were given to volunteers. Organisers purchased these badges from an NGO, who made these badges in the form of a circle inscribed in a square of side 8 cm.

O is the centre of the circle and $∠ A O B = 9 0^{\circ}$ . Based on the above information, answer the following questions:

(i)
What is the area of square ABCD?
1 mark
(ii)
What is the length of diagonal AC of square ABCD?
1 mark
(iii)
Find the area of sector OPRQO.
2 marks
OR
(iii)
Find the area of remaining part of square ABCD when area of circle is excluded.
2 marks

Figure in the paper: Square ABCD with A bottom-left, B bottom-right, C top-right, D top-left. A circle is inscribed in the square, touching all four sides; its centre O is marked in the middle. Side AB = 8 cm, with the point R marked at the bottom where the circle touches AB. Points P and Q lie on the circle near A and B respectively, and radii OP and OQ are drawn with $∠$ AOB = 90° marked at O. Sector OPRQO is shaded/marked by the arc from P through R to Q. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/6/1.

Hints

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

Hint 1

The diameter of the inscribed circle equals the side of the square.

Hint 2

For the diagonal, use Pythagoras theorem in triangle ABC.

Hint 3

Sector area is $\frac{θ}{36 0 ^{\circ}} π r^{2}$ . For the remaining part, subtract the circle's area from the square's area.

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) Area of square ABCD $= side^{2} = 8^{2} = 64 cm^{2}$
1 mark
(ii) By Pythagoras theorem in $△ A B C$ : $A C = 8^{2} + 8^{2} = 128 = 82$ cm
1 mark
(iii)(a) The radius of the inscribed circle is half the side: $r = \frac{8}{2} = 4$ cm, and $θ = 9 0^{\circ}$ .
Area of sector $= \frac{θ}{36 0 ^{\circ}} \times π r^{2} = \frac{90}{360} \times \frac{22}{7} \times 4 \times 4$
1 mark
$= \frac{88}{7} cm^{2}$ (about $12.57 cm^{2}$ )
1 mark
(iii)(b) Area of circle $= π r^{2} = \frac{22}{7} \times 4 \times 4 = \frac{352}{7} cm^{2}$
1 mark
Remaining area $= 64 - \frac{352}{7} = \frac{448 - 352}{7} = \frac{96}{7} cm^{2}$ (about $13.71 cm^{2}$ )
1 mark

Answer: (i) 64 cm²; (ii) 8√2 cm; (iii)(a) 88/7 cm²; (iii)(b) 96/7 cm²

Where the marks are

Area of square = 64 cm² (1 mark)
Diagonal $A C = 82$ cm (1 mark)
(iii)(a) Uses sector formula with r = 4 and θ = 90° (1 mark); gets 88/7 cm² (1 mark)
(iii)(b) Area of circle = 352/7 cm² (1 mark); remaining area = 96/7 cm² (1 mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Sector of a Circle

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 paper2023 board paper, set 430/6/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2023: question 36

Hints

Step-by-step solution

Revise the chapter

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