Now, compute the sector area: \(rac{120}{360} imes 100\pi = rac{1}{3} imes 100\pi = rac{100\pi}{3} pprox 104.72 ext{ cm}^2\).

["# How to Compute Sector Area: A Step-by-Step Guide (Including Example with ( 120^\circ ) Sector)", "Understanding how to compute the area of a circular sector is essential in geometry, trigonometry, engineering, architecture, and data visualization. In this article, we’ll explore the formula for sector area, walk through a clear example involving a ( 120^\circ ) sector, and explain why the result is approximately ( 104.72 , \ ext{cm}^2 ) when using ( \pi \approx 3.1416 ).", "## What Is a Sector?", "A sector is the region enclosed by two radii of a circle and the arc between them. Imagine slicing a pie—each slice is a sector. The size of the sector depends on the central angle in degrees.", "## The Formula for Sector Area", "The area ( A ) of a sector with central angle ( \ heta^\circ ) and radius ( r ) is given by:", "[\nA = \frac{\ heta}{360} \ imes \pi r^2\n]", "### Breakdown of the Formula:\n- ( \ heta ): central angle in degrees (fraction of full circle)\n- ( \pi r^2 ): area of the full circle\n- ( \frac{\ heta}{360} ): fraction of the circle represented by the sector\n- Multiplying gives the portion of the circle’s area", "---", "## Example: Computing Area of a ( 120^\circ ) Sector", "Let’s compute the sector area when ( \ heta = 120^\circ ) and ( r = 10 , \ ext{cm} ) (since ( 100\pi \approx 314.16 ), using radius ( r = 10 ) simplifies calculation while matching the (\pi) in the final formula).", "### Step 1: Plug values into the formula", "[\nA = \frac{120}{360} \ imes \pi \ imes (10)^2\n]", "### Step 2: Simplify", "- ( \frac{120}{360} = \frac{1}{3} )\n- ( 10^2 = 100 )", "[\nA = \frac{1}{3} \ imes \pi \ imes 100 = \frac{100\pi}{3}\n]", "### Step 3: Approximate with ( \pi \approx 3.1416 )", "[\nA \approx \frac{100 \ imes 3.1416}{3} = \frac{314.16}{3} = 104.72\n]", "Thus, the sector area is approximately ( 104.72 , \ ext{cm}^2 ).", "---", "## Why This Matters Beyond Math", "Computing sector areas applies in:", "- Design & Manufacturing: Calculating material usage for curved components\n- Architecture: Estimating roof or arch-shaped space\n- Physics: Modeling rotating parts and angular displacement\n- Computer Graphics: Rendering circular segments in visualizations", "---", "## Key Takeaways", "- A sector’s area depends on radius and central angle\n- Convert degrees to a fraction of a full circle: ( \frac{\ heta}{360} )\n- Using ( r = 10 ) simplifies calculations and keeps the core formula transparent\n- Approximating ( \pi ) enables practical numerical results", "---", "### Summary", "Using the sector area formula:", "[\nA = \frac{\ heta}{360} \ imes \pi r^2\n]", "For a ( 120^\circ ) sector with ( r = 10 , \ ext{cm} ):", "[\nA = \frac{120}{360} \ imes \pi \ imes 100 = \frac{100\pi}{3} \approx 104.72 , \ ext{cm}^2\n]", "Mastering this technique empowers you to solve real-world problems involving curved shapes efficiently.", "---", "Keywords: sector area formula, compute sector area, trigonometry tutorial, circular sector, ( 120^\circ sector calculation, π approximation, geometry examples, area of a circle sector"]