The diameter of a circle is equal to the side of a square. If the area of the circle is 49π square centimetres, what is the area of the square?

Introduction / Context:
This question connects the geometry of a circle and a square. The diameter of the circle is equal to the side length of the square. Given the area of the circle, we use the circle area formula to find the radius and then the diameter. That diameter becomes the side of the square, and from that we compute the square area. This is a straightforward application of area formulas for a circle and a square.

Given Data / Assumptions:
- Area of the circle is 49π square centimetres. - Let radius of the circle be r. - Circle area formula: Area = π * r^2. - Diameter of the circle is 2r, and this equals the side of the square. - We must find the area of the square.

Concept / Approach:
First, use the circle area formula to find r. Once we know r, the diameter is just 2r. Since the side of the square equals the diameter, we can write side = 2r. Then the area of the square is side^2, which is (2r)^2. Substituting the value of r gives the required area of the square. The problem relies on careful use of formulas, not on any advanced geometry.

Step-by-Step Solution:
Step 1: Use circle area formula: Area of circle = π * r^2. Step 2: We are given that π * r^2 = 49π. Step 3: Divide both sides by π to get r^2 = 49. Step 4: Take square root of both sides: r = √49 = 7 cm. Step 5: Diameter of the circle is 2r = 2 * 7 = 14 cm. Step 6: The side of the square is equal to this diameter, so side of square s = 14 cm. Step 7: Area of the square is s^2 = 14^2. Step 8: Compute 14^2 = 196 square centimetres.

Verification / Alternative check:
We can double check by working backward. If the square side is 14 cm, its area is 196 square centimetres. The diameter of the circle is also 14 cm, so radius is 7 cm. The area of the circle with radius 7 cm is π * 7^2 = 49π, which matches the given circle area. This consistent loop confirms that the square area 196 square centimetres is correct.

Why Other Options Are Wrong:
Option 154 square centimetres might come from mixing the circumference formula or incorrectly squaring. Option 124 square centimetres does not correspond to any logical combination of radius or diameter obtained from 49π. Option 98 square centimetres is exactly half of 196 and may come from forgetting that area is side squared and dividing by 2 accidentally.

Common Pitfalls:
Some students confuse area and circumference and may try to use 2πr = 49π, which would be incorrect here since 49π is clearly an area. Another mistake is to forget that the diameter, not the radius, is equal to the side of the square. Always carefully identify which quantity is being equated when problems link two shapes. Keeping symbols like r, diameter, and side distinct in your steps helps avoid confusion.

Final Answer:
The area of the square is 196 square centimetres.