Circumferences of two circles are equal. Is it necessary that their areas are equal? Why?
Step 1: Recall the formula for the circumference of a circle.
Circumference \(C = 2 \pi r\), where \(r\) is the radius (measured in metres in SI).
Step 2: Suppose the circumferences of two circles are equal:
\(2 \pi r_1 = 2 \pi r_2\)
Step 3: Cancel \(2\pi\) from both sides:
\(r_1 = r_2\)
Step 4: If the radii are equal, then both circles are of the same size.
Step 5: Recall the formula for the area of a circle:
Area \(A = \pi r^2\), measured in square metres (m²).
Step 6: Since \(r_1 = r_2\), their areas will also be equal:
\(\pi r_1^2 = \pi r_2^2\)
Final Answer: Yes, if two circles have equal circumferences, their radii are equal, so their areas must also be equal. True.