CBSE Class 10 Mathematics Chapter 11 Areas Related to Circles — 20 MCQs with Answers

Before tackling sectors and segments, ensure you are fluent with the circle's basic formulas. The circumference (perimeter) is 2πr and the area is πr². These serve as building blocks for every other formula in Chapter 11. In the 2024 CBSE paper, 1 MCQ tested whether students could identify the correct area formula when radius was given as a surd. Revise the distinction between diameter and radius, and practice substituting values carefully — a common slip is writing 2r instead of r² in the area formula.

A sector is the region enclosed by two radii and the intercepted arc. The NCERT derives two key formulas: area of sector = (θ/360)×πr² and arc length = (θ/360)×2πr, where θ is the central angle in degrees. When θ = 90°, the sector is a quadrant; θ = 180° gives a semicircle. About 40 per cent of MCQs in this chapter test these formulas directly or ask for the angle when area is known. Always check units: if the question uses π = 22/7, stick with fractions throughout; decimal approximations can introduce rounding errors.

A segment is the region between a chord and the arc it subtends. To find its area, subtract the area of the triangle formed by the two radii and the chord from the area of the corresponding sector. For a minor segment with central angle θ, Area of segment = (θ/360)×πr² − (1/2)r²sinθ (if using trigonometry) or, more commonly in Class 10, Area of segment = Area of sector − Area of triangle. The NCERT example uses a semicircular segment (where the triangle is right-angled), which simplifies to (πr²/2) − (r²/2). About 20 per cent of board MCQs involve segments, often combined with a quadrant or semicircle.

NCERT dedicates several examples to finding areas of shaded regions when a circle is inscribed in or circumscribed about a square or rectangle. A circle inscribed in a square of side a has radius a/2; circumscribed about a square of side a has radius a√2/2. For rectangles with sides l and w, the circumradius is √(l²+w²)/2. These combinations appear in 2-3 MCQs per paper, often requiring you to subtract or add areas. Always sketch a quick diagram and label all known lengths before writing formulas. CBSETUTOR.ai's photo-upload tool can instantly verify your diagram-based solutions and highlight algebraic mistakes in real time.

When a circle is inscribed in an equilateral triangle of side a, the inradius r = a/(2√3); when circumscribed, the circumradius R = a/√3. These formulas are derived in NCERT Exercise 11.2 and are frequently tested in assertion-reason MCQs. A typical question gives the triangle's side and asks for the area of the circle or the shaded region between the triangle and the circle. Remember that the area of an equilateral triangle is (√3/4)a². Practice sketching the triangle, marking the centre, and labeling radii to avoid sign errors when subtracting areas.

CBSE loves composite figures made from semicircles on the sides of a square or quadrants at the corners of a rectangle. A common pattern: a square of side a has semicircles drawn on all four sides outward; the total perimeter then involves four semicircular arcs. Another favourite is four quadrants removed from the corners of a square, leaving a flower-like shape in the centre. These multi-step problems test both formula recall and arithmetic accuracy. Work in fractions (using π = 22/7) throughout, and simplify only at the end to minimize rounding errors. CBSETUTOR.ai's step-by-step solver can walk you through each layer of such combinations, showing where you dropped a factor or miscounted quadrants.

The 2024 CBSE board introduced assertion-reason (A-R) questions in Section A. Each A-R MCQ presents two statements: Assertion (A) and Reason (R). You must decide if both are true, if A is true but R is false, if A is false but R is true, or if both are false, and whether R correctly explains A. In Chapter 11, typical assertions link the area of a sector to the central angle, or claim that the area of a segment is always less than the area of the corresponding sector. The reason often states a formula or a definition. Read both statements independently first, verify their truth, then check the logical link. About 10 per cent of the 20 MCQs in the board paper are A-R format.

Section A of the CBSE Class 10 Mathematics paper carries 20 MCQs for 20 marks, each with four options and no negative marking. Allocate roughly 30–35 minutes to this section, spending about 1.5 minutes per MCQ. Start by scanning all 20 questions and tick those that you can solve in under 30 seconds — usually direct formula-based ones like 'find the area of a circle given radius' or 'arc length with angle and radius'. Solve these first to bank easy marks and build confidence. For multi-step combination problems, draw a quick labelled diagram in the margin; this prevents conceptual errors. If an MCQ involves heavy arithmetic, plug in the answer options backwards (work from the choices to see which satisfies the condition). Always double-check units: if the question gives diameter, halve it before using radius formulas. Finally, review any assertion-reason MCQs last, as they reward careful reading more than calculation speed. CBSETUTOR.ai's unlimited MCQ practice module tracks your speed and accuracy per topic, so you can identify whether you are losing time on segments or combination figures and adjust your revision accordingly.