GATE 2026 EC GA Q.9: solution and answer

GATE 2026 · ECGA Q.9 · 2 marks · MCQ

The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).

The ratio of the surface area of the land to that of the water in the city of Atlantis is ______ (round off to two decimal places).

GATE 2026 EC GA question 9: Six concentric circles around a common centre, marked with a dot labelled "Center". — Figure from the GATE 2026 EC paper. It is part of the question.

(A)
0.45
(B)
0.60
(C)
0.75
(D)
0.90

Answer and solution

ANSWER

The land-to-water area ratio is $\frac{78.25 π}{104 π} \approx 0.75$ , so the answer is option (C).

Official answer: CSource: Official answer key, GATE 2026 EC (IIT Guwahati)

Build the radii outward from the centre by adding the widths (all in stades): Inner Island $r_{A} = 2.5$ ; $r_{B} = 2.5 + 1 = 3.5$ ; $r_{C} = 3.5 + 2 = 5.5$ ; $r_{D} = 5.5 + 2 = 7.5$ ; $r_{E} = 7.5 + 3 = 10.5$ ; $r_{F} = 10.5 + 3 = 13.5$ .
The area of a ring between radii $r_{1} < r_{2}$ is $π (r_{2}^{2} - r_{1}^{2})$ , and the Inner Island is a full disc of area $π r_{A}^{2}$ . Every area carries the factor $π$ , which cancels in the ratio, so work with the $r^{2}$ values only.
Squares of the radii: $2. 5^{2} = 6.25$ , $3. 5^{2} = 12.25$ , $5. 5^{2} = 30.25$ , $7. 5^{2} = 56.25$ , $10. 5^{2} = 110.25$ , $13. 5^{2} = 182.25$ .
Land (shaded: the Inner Island disc, ring BC and ring DE): $6.25 + (30.25 - 12.25) + (110.25 - 56.25) = 6.25 + 18 + 54 = 78.25$
Water (unshaded: rings AB, CD and EF): $(12.25 - 6.25) + (56.25 - 30.25) + (182.25 - 110.25) = 6 + 26 + 72 = 104$
Check: land + water must equal the whole city, a disc of radius $13.5$ : $78.25 + 104 = 182.25 = 13. 5^{2}$ . It does. (Each ring can also be checked as $(r_{2} - r_{1}) (r_{2} + r_{1})$ , e.g. ring DE: $3 \times 18 = 54$ .)
Ratio of land to water: $\frac{78.25}{104} = 0.7524 \dots \approx 0.75$ which is option (C).

KEY IDEA

A ring of inner radius $r_{1}$ and outer radius $r_{2}$ has area $π (r_{2}^{2} - r_{1}^{2})$ . Work out each boundary radius by adding the widths from the centre outward. The factor $π$ cancels in a ratio of areas, so only the differences of squared radii matter. A quick check is that all the pieces together must add up to the area of the full disc.

WATCH OUT FOR

Comparing widths instead of areas: land widths $2.5 + 2 + 3 = 7.5$ against water widths $1 + 2 + 3 = 6$ gives $1.25$ , but outer rings of the same width cover much more area. Another slip is leaving the Inner Island out of the land total, which gives $72/104 \approx 0.69$ . Others stop the city at E and drop the outer water ring EF, or divide land by the whole city ( $78.25/182.25 \approx 0.43$ ) instead of by the water.

From the GATE 2026 EC paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

GATE 2026 EC GA Q.9: solution and answer

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