What is the diameter of the largest circle that can be drawn on a chessboard so that its entire circumference gets covered by the black squares and no part of the circumference falls on any white space, given that the chessboard has black and white squares of side one inch?

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — $10$ inches

Detailed Solution

The circle can pass from one black square to the next only through a shared corner, since black squares touch only diagonally.

Centre the circle at the centre of a black square. The corners at offsets $(\pm 1.5, \pm 0.5)$ and $(\pm 0.5, \pm 1.5)$ are all at distance $1.5^{2} + 0.5^{2} = 2.5$ .

The circle of radius $2.5$ goes through these $8$ corners and between them stays inside black squares, so its diameter is $2 2.5 = 10$ inches.

✓ Check

(\frac{10}{2})^{2} = 2.5 = 1.5^{2} + 0.5^{2}

⚠ Common trap

Options (A) and (B) also fit on black squares (a circle inside one black square; a circle through the four corners of a white square), but they are smaller than

10

Quick tip

Radius through corners $(1.5, 0.5)$ : $r^{2} = 1. 5^{2} + 0. 5^{2} = 2.5 \Rightarrow d = 10$

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Geometry — Preparation

This question tests your understanding of Geometry, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

Geometry in entrance exams covers lines and angles, triangles, quadrilaterals, polygons and circles. Questions test properties of angles formed by parallel lines, congruence and similarity of triangles, the Pythagorean theorem, properties of special quadrilaterals, interior and exterior angles of polygons, and circle theorems such as angles in the same segment and tangent properties. Most geometry questions are solved by recognising the right property, not by long calculation. A neat rough diagram, labelled with the given information, often reveals the answer. Pythagorean triplets (3-4-5, 5-12-13, 8-15-17) and the 30-60-90 and 45-45-90 triangle ratios are worth memorising because they appear repeatedly. Practise the geometry questions below. Each comes with a step-by-step solution that names the property used.

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What is the diameter of the largest circle that can be drawn on a chessboard so that its entire circumference gets covered by the black squares and no part of the circumference falls on any white space, given that the chessboard has black and white squares of side one inch?

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