ORIGINAL · SAMPLE QNS · Q1.1Mathematics › Analytic geometry
What is most nearly the area of the square circumscribed by a circle of area 314 sq. in?
Bare-bones solution
r = √(314 / π) = 10 in
diagonal d = 2r = 20 in
A = d2 / 2 = 400 / 2 = 200 sq. in
Answer: (B) 200 sq. in
References
Common formulas (area of a circle, area of a square, Pythagorean theorem); not necessarily given in the NCEES® Handbook.
Detailed solution
Step 1 · Read the geometry. The square sits inside the circle with its corners on it, so the square's diagonal equals the circle's diameter.
Step 2 · Radius. A = πr2 gives r = √(314 / π) = 10 in, so the diagonal is 2r = 20 in.
Step 3 · Side from Pythagoras. s2 + s2 = 202, so s2 = 400 / 2 = 200. The area is s2 = 200 sq. in.
Sanity check: a square inside a circle covers 2/π ≈ 64% of it, and 0.64 × 314 ≈ 200.
What if I were to guess?
Two options can go without any calculation at all.
- The square sits completely inside the circle, so its area must be less than 314. Cross out
(D) 350 sq. in. - Fold the four circle segments outside the square in across its sides (dotted arcs below). They don't fill the square, so the square is more than half the circle (314 / 2 = 157). Cross out
(A) 150 sq. in.
That leaves (B) or (C), a 50/50 guess without any calculation.
Distractor analysis
(A) 150 sq. in roughly halves the circle's area (314 / 2 ≈ 157).
(C) 250 sq. in overestimates how much of the circle the square fills.
(D) 350 sq. in is larger than the circle itself, which is impossible for a square drawn inside it.