E1 ate) 3. Explain why the area of a circle with radius \( r \) is four times the area of a quarter-circle with radius \( r \), but the circumference of a circle with radius \( r \) is less than four times the circumference of a quarter-circle with radius \( r \).

The area of a full circle is given by the formula \( A = \pi r^2 \). A quarter-circle, being one fourth of the full circle, has an area of \( A_{quarter} = \frac{1}{4} \pi r^2 \). When you multiply the area of a quarter-circle by four, you get \( 4 \times A_{quarter} = 4 \times \left(\frac{1}{4} \pi r^2\right) = \pi r^2 \), which is indeed the area of the full circle. On the other hand, the circumference of a circle is calculated using \( C = 2\pi r \). The circumference of a quarter-circle (which includes just one of the four curves) is \( C_{quarter} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \). Therefore, four times the circumference of a quarter-circle is \( 4 \times C_{quarter} = 4 \times \left(\frac{\pi r}{2}\right) = 2\pi r \), which is not equal to the circumference of the full circle \( C = 2\pi r \), but matches perfectly. It’s all about the elegant relationship between area and circumference in the world of circles!