Circle Geometry

The largest circle that fits inside a square is the inscribed circle, whose diameter equals the side of the square. With side 14m, radius = 7m. Area = πr² = π × 49 ≈ 154 sq m.

The formula for the area of a circle is $\pi r^2$. The expression $2\pi r$ represents the circumference of a circle, making this statement incorrect.

The area of a circle is pi * R^2. Using 22/7 as the standard approximation for pi, the formula is (22/7) * R^2. Option 407792 is the correct representation of this area formula.

The perimeter (circumference) of a circle is 2 * pi * R. Using 22/7 as an approximation for pi, the formula becomes 2 * (22/7) * R. Option 407798 correctly identifies this geometric formula.

The largest square inscribed in a circle has a diagonal equal to the circle's diameter, which is 8 cm. The area of a square with diagonal $d$ is $d^2 / 2$. Thus, the area is $8^2 / 2 = 64 / 2 = 32$ square centimeters. Distractors represent incorrect formulas or area of the circumscribed square (64).

A square circumscribing a circle of diameter 8 cm has a side length equal to the diameter of the circle, which is 8 cm. The perimeter of this square is $4 \times 8 = 32$ cm. Distractors like 16, 24, and 48 represent incorrect side-to-perimeter scaling or area calculations.

Area is proportional to the square of the radius. If the radius becomes 0.9r, the area becomes (0.9)^2 = 0.81 of the original area. The diminution is 1 - 0.81 = 0.19, or 19%.

Area is proportional to the square of the radius. If the radius becomes 0.9r, the area becomes (0.9)^2 = 0.81 of the original area. The decrease is 1 - 0.81 = 0.19, or 19%.

The area of a full circle is πR². A quadrant is one-fourth of a circle, so its area is (πR²)/4.

In mathematics, any finite number multiplied by zero results in zero. Regardless of the complexity of the areas calculated in the first two parts of the expression, the final multiplication by zero makes the product 0.

The formal mathematical definition of pi is the ratio of a circle's circumference (C) to its diameter (D), written as pi = C/D. Option C incorrectly reverses this ratio (diameter to circumference would give 1/pi), while option E incorrectly uses radius instead of diameter (C/radius would equal 2pi, not pi). Option B is only a decimal approximation.

The largest circle that can be circumscribed in a rectangle must fit within the smaller dimension. For a 24x88 rectangle, the maximum diameter equals the smaller side (24), as the circle cannot exceed either dimension.

For a square inside a circle, the diagonal equals the diameter. Diagonal = 2 * 5*sqrt(2) = 10*sqrt(2). Side = diagonal/sqrt(2) = 10. Area = 10^2 = 100 sq units.

A square inscribed in a circle has a diagonal equal to the circle's diameter. With radius 5, the diameter is 10. The area of a square is (diagonal^2)/2, so (10^2)/2 = 100/2 = 50 square units.

Area depends on the square of the radius (A = πr²). When radius becomes 0.9r (10% reduction), new area = π(0.9r)² = 0.81πr². The reduction is 1 - 0.81 = 0.19 = 19%. Option A (10%) would be true for circumference, not area. Option C (20%) ignores the squared relationship.