AB and CD are diameters of a circle with centre O and radius 7 cm. If ∠BOD = 30°, then find the area and perimeter of the shaded region.

Area =77/3 cm² ≈ 25.67 cm²; Perimeter =106/3 cm ≈ 35.33 cm

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2025: question 29

Set in the CBSE Class 10 board exam of 2025, paper 430/6/1, section C, question 29, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29 · 3 marks · Short answerAreas Related to Circles

AB and CD are diameters of a circle with centre O and radius 7 cm. If $∠$ BOD = $3 0^{\circ}$ , then find the area and perimeter of the shaded region.

Figure in the paper: Circle with centre O and two diameters AB and CD. Points D and B are on the upper right, A and C on the lower left. The shaded regions are the two segments-like triangular regions between chords: one between D, B and the arc near the upper right, and one between A, C and the arc near the lower left. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/6/1.

Hints

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

Hint 1

The two shaded parts are sectors; what is ∠AOC compared with ∠BOD?

Hint 2

Area of a sector is $\frac{θ}{360} π r^{2}$ ; its perimeter is two radii plus the arc $\frac{θ}{360} 2 π r$ .

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.

$∠ A O C = ∠ B O D = 3 0^{\circ}$ [vertically opposite angles], so the shaded region consists of two equal sectors of radius 7 cm and angle $3 0^{\circ}$ .
Area $= 2 (\frac{θ}{36 0 ^{\circ}} \times π r^{2}) = 2 (\frac{30}{360} \times \frac{22}{7} \times 7 \times 7)$
1 mark
$= 2 \times \frac{77}{6} = \frac{77}{3} cm^{2} \approx 25.67 cm^{2}$
½ mark
Perimeter of each sector $= 2 r + \frac{θ}{36 0 ^{\circ}} \times 2 π r$ .
Perimeter of shaded region $= 2 (14 + \frac{30}{360} \times 2 \times \frac{22}{7} \times 7) = 2 (14 + \frac{11}{3})$
1 mark
$= 2 \times \frac{53}{3} = \frac{106}{3} cm \approx 35.33 cm$
½ mark

Answer: Area $= \frac{77}{3}$ cm² ≈ 25.67 cm²; Perimeter $= \frac{106}{3}$ cm ≈ 35.33 cm

Where the marks are

Uses area of two sectors of angle 30° correctly (1 mark)
Area $= \frac{77}{3}$ cm² (½ mark)
Perimeter expression $2 (14 + arc length)$ (1 mark)
Perimeter $= \frac{106}{3}$ cm (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Shaded Region Problems with Sectors

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 430/6/1 (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 2025: question 29

Hints

Step-by-step solution

Revise the chapter

More questions from Areas Related to Circles