Solution: The area of a circular sector is \( A = \frac{\theta}{360} \pi r^2 \). Original area: \( \frac{90}{360} \pi (12)^2 = 36\pi \). New radius: \( 12 + 4 = 16 \) meters. New area: \( \frac{90}{360} \pi (16)^2 = 64\pi \). The increase is \( 64\pi - 36\pi = 28\pi \). \boxed{28\pi}

["Understanding Circular Sectors: Calculating Area Changes with Radius", "When analyzing shapes in geometry, circular sectors often play a key role in engineering, architecture, and design. One common application involves calculating the area of a sector and understanding how changes in radius affect this area—especially when the central angle remains constant.", "### What is the Area of a Circular Sector?", "The area ( A ) of a circular sector with central angle ( \ heta ) degrees and radius ( r ) is given by the formula:\n[\nA = \frac{\ heta}{360} \pi r^2\n]\nThis formula reflects a proportional division of the full circle’s area (( \pi r^2 )) based on the fraction ( \frac{\ heta}{360} ) of the complete 360° circle.", "### Calculating the Original and New Sector Area", "Consider a sector with a central angle of ( 90^\circ ) and initial radius ( r = 12 ) meters:\n[\nA_{\ ext{original}} = \frac{90}{360} \pi (12)^2 = \frac{1}{4} \pi (144) = 36\pi \ ext{ square meters}\n]", "The radius is increased by 4 meters: new radius ( r = 16 ) meters. The sector’s area becomes:\n[\nA_{\ ext{new}} = \frac{90}{360} \pi (16)^2 = \frac{1}{4} \pi (256) = 64\pi \ ext{ square meters}\n]", "The increase in area is:\n[\n\Delta A = A_{\ ext{new}} - A_{\ ext{original}} = 64\pi - 36\pi = 28\pi \ ext{ square meters}\n]", "### Why This Matters", "This calculation helps visualize how modifying dimensions impacts area—a fundamental concept in design, land measurement, and fluid dynamics. Even small radius changes can yield significant area differences, especially with larger circles or narrow central angles.", "### Takeaways", "- Circular sector area depends linearly on radius squared.\n- Fixed-angle sectors expand proportionally when radius increases.\n- Knowing these relationships enables precise calculations in real-world applications.", "The increase in area is:\n[\n\boxed{28\pi}\n]\nsquare meters.", "---", "This SEO-optimized article targets keywords like “circular sector area formula,” “how radius affects sector area,” and “calculate increase in circular sector,” improving search visibility for students, educators, and professionals seeking clear geometry explanations."]