A circular logo of radius 14 cm is fitted perfectly inside a square plaque. Find the area of unused square space around the circle.

Correct — A. A circle fitted perfectly inside a square touches all four sides, so the square's side = diameter.

Side: 2 × 14 = 28 cm
Square area: 28 × 28 = 784 cm²

Circle area: π × 14² = 196π
With π ≈ 3.1416: 196 × 3.1416 ≈ 615.75 cm²

Unused space: 784 − 615.75 = 168.25 cm² → option (a)

Why the others are wrong

• (b)
  56.78 cm² — 56.78 cm² leaves 784 − 56.78 = 727.22 cm² for the circle, which would need π ≈ 3.71. With any usual π the corners total about 168 cm².
• (c)
  161.54 cm² — 161.54 cm² leaves 622.46 cm² for the circle, which would need π ≈ 3.18. With 22⁄7 the circle is 616 cm², leaving 168 cm².
• (d)
  64.26 cm² — 64.26 cm² leaves 719.74 cm² for the circle, which would need π ≈ 3.67, far above any value used for π.

The unused space is the four corners between the square and the circle:
square − circle = (2r)² − πr² = r²(4 − π)

With r = 14 that is 196 × (4 − π) ≈ 196 × 0.8584 ≈ 168.25 cm², about 21.5% of the square, since 1 − π⁄4 ≈ 0.2146.

Key facts

• A circle inscribed in a square has a diameter equal to the square's side.
• Square minus inscribed circle = r²(4 − π) ≈ 0.8584r².
• With π = 22⁄7, a circle of radius 14 cm has area 616 cm².

Common traps

• Taking the square's side as 14 cm, the radius: 196 − 616 is negative, a sign the side should be 28 cm.
• Rejecting 168.25 because 22⁄7 gives exactly 168: the stem fixes no π, so take the nearest option.

No directly related past PYQ was found.