A chord of a circle of radius 10 cm subtends an angle of 60° at the centre O. Find the area of the shaded region. (Use √(3) = 1.73, √(2) = 1.41 and π = 3.14)

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2025: question 30

Set in the CBSE Class 10 board exam of 2025, paper 430/4/1, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerAreas Related to Circles

A chord of a circle of radius 10 cm subtends an angle of $6 0^{\circ}$ at the centre O. Find the area of the shaded region. (Use $3 = 1.73$ , $2 = 1.41$ and $π = 3.14$ )

Figure in the paper: A circle with centre O and a chord AB. Angle AOB = 60° is marked at O. The minor segment between chord AB and the arc is shaded. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/4/1.

Hints

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

Hint 1

Shaded segment = sector − triangle.

Hint 2

With OA = OB and a $6 0^{\circ}$ angle between them, what type of triangle is OAB?

Hint 3

Use $\frac{θ}{360} π r^{2}$ for the sector and $\frac{3}{4} a^{2}$ for the triangle.

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.

Area of sector OAB $= \frac{θ}{36 0 ^{\circ}} \times π r^{2} = \frac{60}{360} \times 3.14 \times 1 0^{2} = \frac{314}{6} \approx 52.33$ cm²
1 mark
OA = OB = 10 cm and $∠ A O B = 6 0^{\circ}$ , so $Δ O A B$ is equilateral. Area of $Δ O A B = \frac{3}{4} \times 1 0^{2} = \frac{1.73 \times 100}{4} = \frac{173}{4} = 43.25$ cm²
1 mark
Area of shaded segment $= \frac{314}{6} - \frac{173}{4} = \frac{628 - 519}{12} = \frac{109}{12} \approx 9.08$ cm²
1 mark

Answer: $\frac{109}{12}$ cm² $\approx 9.08$ cm²

Where the marks are

Area of sector $= \frac{314}{6}$ cm² (1 mark)
Area of equilateral $Δ O A B = \frac{173}{4}$ cm² (1 mark)
Shaded area $= 9.08$ 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 Segment 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 paper2025 board paper, set 430/4/1 (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 2025: question 30

Hints

Step-by-step solution

Revise the chapter

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