Chord AB of a circle with centre O and radius 21 mm subtends an angle of 120° at the centre. Find the perimeters of the shaded region. (Use √(3) = 1.73)

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2026: question 27

Set in the CBSE Class 10 board exam of 2026, paper 30/4/1, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerAreas Related to Circles

Chord AB of a circle with centre O and radius 21 mm subtends an angle of $12 0^{\circ}$ at the centre. Find the perimeters of the shaded region. (Use $3$ = 1.73)

Figure in the paper: Circle with centre O; radii OA and OB with angle AOB = 120°; chord AB drawn; the minor segment between chord AB and the arc AB is shaded. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/4/1.

Hints

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

Hint 1

The boundary of the shaded segment is made of the chord and the arc.

Hint 2

Drop a perpendicular from O to AB; it bisects the 120° angle. Use $sin 6 0^{\circ}$ for half the chord and $\frac{θ}{360} \times 2 π r$ for the arc.

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.

The perimeter of the shaded segment = length of chord AB + length of minor arc AB. Draw $O D ⊥ A B$ . The perpendicular from the centre bisects the chord and $∠ A O B$ , so $∠ A O D = 6 0^{\circ}$ . In right $△ O D A$ : $sin 6 0^{\circ} = \frac{A D}{O A} \Rightarrow \frac{3}{2} = \frac{A D}{21} \Rightarrow A D = \frac{21 3}{2}$ mm.
1 mark
$A B = 2 A D = 213 = 21 \times 1.73 = 36.33$ mm.
½ mark
Length of minor arc AB $= \frac{θ}{36 0 ^{\circ}} \times 2 π r = \frac{120}{360} \times 2 \times \frac{22}{7} \times 21 = 44$ mm.
1 mark
Perimeter of the shaded region $= 44 + 36.33 = 80.33$ mm.
½ mark

Answer: 80.33 mm

Where the marks are

Draws the perpendicular from O and finds $A D = \frac{21 3}{2}$ mm using $sin 6 0^{\circ}$ (1 mark)
Gets chord $A B = 213$ mm (½ mark)
Finds arc length = 44 mm (1 mark)
Adds to get perimeter 80.33 mm (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 paper2026 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2026: question 27

Hints

Step-by-step solution

Revise the chapter

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