Enforcement Officer / Accounts Officer (EO/AO) 2017 recruitment test. Transcribed from the Commission's own booklet (series A, GRS-A-TNS); answer as per the official UPSC final answer key. This question is in Part B. Answer last reviewed 24 August 2026.

The circumference of a circle is 2π cm. Then the area of a square inscribed in the circle is

(a)π/2 cm²
(b)1 cm²
(c)2π cm²
(d)2 cm²

Answer

Why

Correct — D, (d) 2 cm². A square inscribed in a circle has all four corners on the circle, and its diagonal is therefore a diameter — it joins two points of the circle through the centre. That is the step which decides the item, and everything else follows from it.

The circumference is 2π cm, and circumference = 2πr, so the radius is 1 cm and the diameter is 2 cm. The diagonal of the inscribed square is that diameter, 2 cm. For a square of side s the diagonal is s√2, so s√2 = 2 and s = 2 ÷ √2 = √2 cm. The area is s² = (√2)² = 2 cm², which is option (d).

The same result comes out even faster from the halves of the square. The two diagonals of a square are equal and cut each other at right angles, so a square of diagonal d is a rhombus of equal diagonals and its area is half the product of the diagonals, d² ÷ 2. With d = 2 that is 4 ÷ 2 = 2 cm². It is worth keeping the general form: a square inscribed in a circle of radius r has diagonal 2r and area 2r², a relation that answers this whole family of questions in one line.

Two checks confirm the answer. The circle's own area is πr² = π ≈ 3·14 cm², and the square lies inside it, so the square's area must be less than that — as 2 is. In fact the inscribed square always occupies 2 ÷ π, about 64 per cent, of its circle. And the answer contains no π, which is right: the area of the square depends only on r², so π has no way of entering it.

Why the others are wrong

(a)π/2 cm² — The paper prints this option as a two-storey fraction, π over 2, followed by the squared unit; no solidus appears on the page. Its value is about 1·57 cm², which is exactly half the area of the circle, and that is where it comes from — a candidate who assumes that the inscribed square takes up half the circle. It does not: the square occupies 2 ÷ π of the circle, which is about 64 per cent, so half is a clear underestimate. The other reason to distrust the option is the π in it. The area of the square is the square of its side, and the side here comes from the diameter by a division by √2, so the only constants that can appear are rational multiples of r². A π surviving into the answer means the circle's area formula has been used where the square's should have been.
(b)1 cm² — One square centimetre means a side of 1 cm, which is the radius of the circle rather than the side of the square. The confusion is between the radius, the diameter and the diagonal, and it is the characteristic error of the topic: the radius is 1 cm, the diameter and the diagonal are both 2 cm, and the side is 2 ÷ √2 = √2 cm, about 1·41 cm. A square of side 1 cm would fit inside the circle comfortably, with its corners well short of the circumference, so it is not the inscribed square at all. Sketching the figure roughly and labelling the radius, the diameter and the diagonal before any calculation removes this error entirely, and it takes only a few seconds even on a paper where time is short.
(c)2π cm² — 2π cm² is about 6·28 cm², and the circle itself has area πr² = π ≈ 3·14 cm². A square lying inside the circle cannot have twice the circle's area, so this option can be rejected on sight without any calculation at all — which makes it the most useful option in the set for teaching a habit. It is offered because 2π is the number printed in the stem as the circumference, and a candidate under time pressure may carry a familiar expression across from one quantity to another. Circumference and area are different kinds of quantity, measured in different units, and a length can never be transplanted into an answer wanted as an area. Comparing an answer against an obvious bound — here, the area of the figure that contains it — is the cheapest check in geometry.

Concept

A square inscribed in a circle and a circle circumscribed about a square are the same configuration seen from two sides, and one relation governs both: the diagonal of the square is the diameter of the circle. From it everything else follows. If the circle has radius r, the diagonal of the square is 2r, the side is 2r ÷ √2 = r√2, and the area is 2r². If instead the square has side s, the diagonal is s√2, the circle's radius is s ÷ √2 and its area is πs² ÷ 2. The ratio of the square's area to the circle's is always 2r² ÷ πr² = 2 ÷ π, about 0·637, whatever the size — a useful figure for checking any answer in this family. The companion configuration is the circle inscribed in a square, where the circle's diameter equals the side rather than the diagonal, and the ratio of areas is π ÷ 4, about 0·785. Keeping the two apart is a matter of asking which figure is inside: the inner figure's extreme dimension is fixed by the outer figure's, so an inscribed square is limited by the diagonal it can stretch across, while an inscribed circle is limited by the side it must fit between.

This paper closes on the same relation it used at question 117, taken in the opposite direction: there the square was given and the smallest circle around it wanted, here the circle is given and the square inside it wanted. A candidate who has learnt the configuration as a picture rather than as a formula answers both from the same line — diagonal equals diameter — and a candidate who has memorised one direction may not recognise the other. That is exactly what the Commission is testing when it places the pair a few items apart. The stem also makes its first demand before any geometry begins, by giving the circle through its circumference rather than its radius. Converting a given quantity into the one the figure actually needs is the recurring first step in mensuration, and a candidate who omits it here will be working with 2π where 1 belongs.

Key facts

A square inscribed in a circle has all four corners on the circle, so its diagonal is a diameter.
Circumference = 2πr, so a circumference of 2π cm gives a radius of 1 cm and a diameter of 2 cm.
The diagonal of a square of side s is s√2, so a diagonal of 2 cm gives a side of √2 cm.
The area of the square is (√2)² = 2 cm²; in general a square inscribed in a circle of radius r has area 2r².
The area of a square can also be found as half the product of its diagonals, which here is 4 ÷ 2 = 2 cm².
The circle's own area is π ≈ 3·14 cm², so the inscribed square's area must be smaller, and it is 2 ÷ π of it, about 64 per cent.
The area of the square contains no π, because it depends only on the square of the radius.
In the companion configuration — a circle inscribed in a square — the circle's diameter equals the side, not the diagonal.

The diagonal of the inscribed square is the diameter of the circle

Circumference 2π = 2πr, so r = 1 cm and the diameter is 2 cm→A square inscribed in a circle has all four corners on the circle, so its diagonal joins two points of the circle through the centre — it is a diameter. That is the step which decides the item, and everything else follows from it.

s√2 = 2, so s = √2 cm and the area is 2 cm²→In general a square inscribed in a circle of radius r has diagonal 2r and area 2r² — a relation that answers this whole family of questions in one line.

The faster route→The two diagonals of a square are equal and cut each other at right angles, so its area is half the product of the diagonals, d² ÷ 2. With d = 2 that is 4 ÷ 2 = 2 cm².

(c) 2π cm² — rejectable on sight→About 6·28 cm², while the circle itself has area πr² ≈ 3·14 cm². A square lying inside the circle cannot have twice the circle's area. It is offered because 2π is the number printed in the stem as the circumference, and a length can never be transplanted into an answer wanted as an area.

(a) π/2 cm² and (b) 1 cm²→π over 2 — the paper prints it as a two-storey fraction, with no solidus on the page — is about 1·57, exactly half the circle's area, from assuming the inscribed square takes up half the circle; it takes 2 ÷ π, about 64 per cent. A π surviving into the answer means the circle's area formula has been used where the square's should have been. And 1 cm² means a side of 1 cm, the radius mistaken for the side: the radius is 1, the diameter and diagonal are both 2, and the side is √2, about 1·41.

The companion configuration→A circle inscribed in a square has its diameter equal to the side rather than the diagonal, and the ratio of areas is π ÷ 4, about 0·785. Ask which figure is inside: an inscribed square is limited by the diagonal it can stretch across, an inscribed circle by the side it must fit between.

Study next

Squares and circles inscribed in one another, with the ratio of areas in each configurationCircumference, diameter and area of a circle, and converting between them before starting a problemThe diagonal of a square and of a rectangle by the Pythagorean theoremAreas of shaded regions between an inscribed and a circumscribed figureRegular polygons inscribed in a circle, and how the side relates to the radius as the number of sides grows

Common traps

Taking the side of the square to be the diameter, when it is the diagonal that equals the diameter.
Using the radius as the side, which gives an area of 1 rather than 2 square centimetres.
Assuming the inscribed square is half the circle in area; it is about 64 per cent of it.
Carrying π into the answer, when the area of the square depends only on the square of the radius.
Using the given circumference as though it were a radius or an area, without converting it first.

The square-and-circle configuration is one of the most frequently set pieces of geometry in these papers, and the examiner varies it by choosing which quantity is given and which is wanted. The circle may be given by its radius, its diameter, its circumference or its area, and the square may be inside or outside; the question may ask for the side, the area, the perimeter, or the area of the region between the two figures. Every version reduces to one of two relations — diagonal equals diameter for an inscribed square, side equals diameter for an inscribed circle — combined with the conversions among the circle's own measurements. A candidate who writes those two relations at the top of the rough sheet before starting the section will answer this family without hesitation, and will also spot the options that are impossible on size grounds alone.

Related PYQs

EPFO_EOAO_2017_Q117Open & attempt →

The area of the smallest circle which contains a square of area 4 cm² inside is

(a) π cm²
(b) 2π cm²
(c) 3π cm²
(d) 4π cm²

Answer(b) 2π cm²

The same relation taken the other way three items earlier in this paper — a square of given area and the smallest circle that contains it.

EPFO_APFC_2016_Q36

There are two circles of radii r1 and r2 (r1 < r2). The area of the bigger circle is 693/2 cm2. The difference of their circumferences is 22 cm. What is the sum of the diameters of the two circles ?

(a) 17·5 cm
(b) 22 cm
(c) 28·5 cm
(d) 35 cm

Answer(d) 35 cm

Moving between a circle's measurements, which is the first step here: two circles given by an area and by a difference of circumferences, with the sum of the diameters wanted.

Practice

practice — not a real PYQ
The diameter of a circle is 10 cm. What is the area of the largest square that can be inscribed in this circle?
- (a)25 cm²
- (b)50 cm²
- (c)75 cm²
- (d)100 cm²
Answer(b) 50 cm²
practice — not a real PYQ
The area of a square inscribed in a circle of radius r is
- (a)r²
- (b)2r²
- (c)4r²
- (d)πr² ÷ 2
Answer(b) 2r²

Why