If the perimeter of a circle is equal to that of a square, then the ratio of their areas (circle : square) is
\(22:7\)
\(14:11\)
\(7:22\)
\(11:14\)
Step 1: Write the formula for perimeter (circumference) of a circle.
Perimeter of circle = \(2 \pi r\), where \(r\) is the radius (SI unit: metre).
Step 2: Write the formula for perimeter of a square.
Perimeter of square = \(4a\), where \(a\) is the side length (SI unit: metre).
Step 3: Since both perimeters are equal:
\(2 \pi r = 4a\)
Divide both sides by 4:
\(a = \dfrac{\pi r}{2}\)
Step 4: Write the formula for area of a circle.
Area of circle = \(\pi r^2\) (SI unit: m²).
Step 5: Write the formula for area of a square.
Area of square = \(a^2\).
Step 6: Substitute \(a = \dfrac{\pi r}{2}\) into the area of the square.
Area of square = \(\left( \dfrac{\pi r}{2} \right)^2 = \dfrac{\pi^2 r^2}{4}\).
Step 7: Now find the ratio of the areas (circle : square).
\(\dfrac{\text{Area of circle}}{\text{Area of square}} = \dfrac{\pi r^2}{\dfrac{\pi^2 r^2}{4}}\)
Simplify:
\(= \dfrac{\pi r^2 \times 4}{\pi^2 r^2} = \dfrac{4}{\pi}\)
Step 8: Approximate the value of \(\pi\).
\(\pi \approx 3.14\), so \(\dfrac{4}{\pi} \approx \dfrac{4}{3.14} \approx 1.27\).
As a ratio, \(1.27 \approx \dfrac{14}{11}\).
Final Answer: The ratio of their areas is \(14:11\). Hence option (B).