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 \( r \).

To understand the relationship between the area and circumference of a circle and a quarter-circle, let's break it down step by step. ### Area of a Circle vs. Area of a Quarter-Circle 1. **Area of a Circle**: The formula for the area \( A \) of a circle with radius \( r \) is given by: \[ A = \pi r^2 \] 2. **Area of a Quarter-Circle**: The area of a quarter-circle (which is one-fourth of a full circle) is: \[ A_{quarter} = \frac{1}{4} \pi r^2 \] 3. **Comparison of Areas**: If we compare the area of the full circle to the area of the quarter-circle: \[ A = 4 \times A_{quarter} \] Substituting the area of the quarter-circle: \[ A = 4 \times \left(\frac{1}{4} \pi r^2\right) = \pi r^2 \] This shows that the area of the circle is indeed four times the area of the quarter-circle. ### Circumference of a Circle vs. Circumference of a Quarter-Circle 1. **Circumference of a Circle**: The formula for the circumference \( C \) of a circle with radius \( r \) is: \[ C = 2\pi r \] 2. **Circumference of a Quarter-Circle**: The circumference of a quarter-circle includes the arc length plus the two straight edges (the radius lines). The arc length of a quarter-circle is: \[ C_{arc} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \] Therefore, the total circumference of the quarter-circle is: \[ C_{quarter} = C_{arc} + 2r = \frac{\pi r}{2} + 2r \] 3. **Comparison of Circumferences**: Now, let's compare the circumference of the full circle to four times the circumference of the quarter-circle: \[ 4 \times C_{quarter} = 4 \left(\frac{\pi r}{2} + 2r\right) = 2\pi r + 8r \] The circumference of the full circle is \( 2\pi r \), which is less than \( 2\pi r + 8r \). ### Conclusion - The area of a circle is four times the area of a quarter-circle because the area scales with the square of the radius, and a quarter-circle is one-fourth of the full circle. - The circumference of a circle is less than four times the circumference of a quarter-circle because the circumference is a linear measure, and the additional straight edges of the quarter-circle contribute to its total length, making it larger than just the arc length alone. Thus, we see a clear distinction in how area and circumference relate to the shapes in question.