Q.3. If the circumference of a circle and the perimeter of a square are equal, then(a) Area of the circle = Area of the square(b) Area of the circle > Area of the square(c) Area of the circle < Area of the square(d) Nothing definite can be said about the relation between the areas of the circle and square.
Question
Q.3. If the circumference of a circle and the perimeter of a square are equal, then(a) Area of the circle = Area of the square(b) Area of the circle > Area of the square(c) Area of the circle < Area of the square(d) Nothing definite can be said about the relation between the areas of the circle and square.
Solution
The answer is (b) Area of the circle > Area of the square.
Here's the step-by-step solution:
Step 1: Let's denote the circumference of the circle as C and the perimeter of the square as P. Given that C = P, we can write the formulas for both as follows:
C = 2πr (where r is the radius of the circle) P = 4s (where s is the side of the square)
Step 2: Since C = P, we can equate the two formulas:
2πr = 4s => r = 2s/π
Step 3: Now, let's compare the areas of the circle and the square. The area of the circle is πr² and the area of the square is s². Substituting r from step 2:
Area of the circle = π(2s/π)² = 4s²/π Area of the square = s²
Step 4: Comparing the two areas, we can see that the area of the circle is greater than the area of the square if π < 4. Since π is approximately 3.14, this condition is true. Therefore, the area of the circle is greater than the area of the square.
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