Circle Geometry

Area R = πr², S = πs², T = πt². Given R=2S, then πr²=2πs² => r=s√2. Given S=2T, then πs²=2πt² => s=t√2. Substituting s: r = (t√2)√2 = 2t. Thus, r = 2t. The statement 'Col A is not equal to Col B' is False because they are equal.

The Ashoka Chakra's radius is exactly 3/8th of the width of the white strip in the Indian flag, as specified in the Flag Code of India. This proportion ensures the chakra is centered and appropriately sized within the white band.

The area of a circle is calculated using the formula A = πr², where π (pi) is approximately 3.14159 and r is the radius of the circle. Option B '2πr' gives the circumference, not area. Option A '½ base × height' is for triangles. Option C 'x + y' is a simple algebraic expression with no geometric meaning. The area formula πr² is fundamental to geometry.

A semicircle's perimeter is (π * r) + 2r. For r=2: (3.14 * 2) + (2 * 2) = 6.28 + 4 = 10.28. 6.28 is just the arc length, and 12.56 is the perimeter of the original full circle.

A circle has infinite diameters because any line passing through the center to any point on the circumference is a diameter. Since there are infinite points on a circle's circumference, there are infinite possible diameters. This is a fundamental property of circles.

The area of a circle is πr², where r is the radius. Option D 'Pi r squared' correctly expresses this formula. Options A and B don't include the radius variable properly, and option C (2πr) is actually the formula for circumference, not area. The area formula can be derived by dividing a circle into infinitesimal sectors and rearranging them into a rectangle.

When the radius increases by 6%, the new radius becomes 1.06r. Area is proportional to the square of the radius (A = πr²), so the new area becomes π(1.06r)² = 1.1236πr². This represents a 12.36% increase over the original area. The percentage increase is calculated as (1.06² - 1) × 100 = 12.36%. This demonstrates how small percentage changes in radius compound when applied to area.

Pi is the correct answer - it is the mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The golden ratio is a different constant (about 1.618), while nu and xi are other Greek letters with no connection to this ratio.

For a square of side 6cm, the inscribed circle has diameter 6cm (radius 3cm). Circle area = πr² = 9π. Square area = 6² = 36. The ratio is 9π/36 = π/4 ≈ 0.785. Among options, 11/14 ≈ 0.786 is the closest approximation to π/4.

If diameter increases by 100%, it doubles (2x). Since area is proportional to the square of the diameter (or radius), the new area is 2 squared, or 4 times the original. A 4x increase in area represents a 300% increase (400% - 100% = 300%).

The largest square inside a circle has its diagonal equal to the circle's diameter. For radius 10 cm, diameter = 20 cm. If diagonal = 20, then side = diagonal/√2 = 20/√2 = 10√2. Area = side² = (10√2)² = 200 sq cm. The claimed answer B is correct. Option C (100) would be if side = 10, but the diagonal constraint makes the square larger.

The largest circle inside a square has its diameter equal to the square's side. For side 14 m, diameter = 14 m, so radius = 7 m. Area = πr² = π × 7² = π × 49 = 154 sq m (using π ≈ 22/7). The claimed answer B is correct. Option D has the right number but wrong units (cm instead of m).

The circle's diameter is x - 6, and this diameter must be less than 3 (and positive). So 0 < x - 6 < 3, meaning 6 < x < 9. Only x = 8 satisfies this condition and is among the options (2x-8 = 8 when x = 8). x^2 gives values 36-81, 10 is outside the range, and x+1 cannot equal x.

The diameter is (x-6) and must be less than 3, so x < 9. Setting x = 2x-8 gives x = 8, making the diameter 2, which satisfies the condition. Option A (x = x² gives x = 0 or 1) leads to negative diameters, option B (x = 10) gives diameter 4 > 3, and option C (x = x+1) is impossible.