Question:** A regular hexagon is inscribed in a circle with a radius of 10 cm. Calculate the area of the hexagon.
["# Calculate the Area of a Regular Hexagon Inscribed in a Circle with Radius 10 cm", "When it comes to geometric shapes, the regular hexagon stands out for its symmetry and elegant mathematical properties—especially when inscribed in a circle. If you're asked to calculate the area of a regular hexagon with a circumscribed circle (circumradius) of 10 cm, understanding the formula and derivation is key to mastering this concept.", "## Why a Regular Hexagon in a Circle Matters", "A regular hexagon inscribed in a circle has all six of its vertices touching the circle's perimeter, meaning the circle’s radius equals the distance from the center to each vertex. This unique relationship simplifies area calculations significantly. In fact, a regular hexagon can be divided into 6 identical equilateral triangles, each with side length equal to the circle’s radius.", "## Step-by-Step to Compute the Area", "### Step 1: Understand the Structure", "- A regular hexagon inscribed in a circle with radius ( r = 10 ) cm\n- The hexagon is made of 6 equilateral triangles\n- Each triangle has two sides equal to the radius (10 cm), and the included angle is ( 60^\circ ), confirming an equilateral shape", "### Step 2: Area of One Equilateral Triangle", "The formula for the area of an equilateral triangle with side length ( s ) is:", "[\n\ ext{Area} = \frac{\sqrt{3}}{4} s^2\n]", "Since each side of the triangle is equal to the radius:\n( s = 10 ) cm", "[\n\ ext{Area of one triangle} = \frac{\sqrt{3}}{4} \ imes 10^2 = \frac{\sqrt{3}}{4} \ imes 100 = 25\sqrt{3} \ ext{ cm}^2\n]", "### Step 3: Multiply by 6 for the Full Hexagon", "[\n\ ext{Total area} = 6 \ imes 25\sqrt{3} = 150\sqrt{3} \ ext{ cm}^2\n]", "### Final Answer:", "[\n\boxed{150\sqrt{3} \ ext{ cm}^2}\n]", "## Conclusion", "Calculating the area of a regular hexagon inscribed in a circle is straightforward when recognizing its broken-down triangular structure. With a radius of 10 cm, the hexagon’s area is ( 150\sqrt{3} ) square centimeters—showcasing a blend of geometric beauty and precise mathematical computation.", "---", "### Key SEO Keywords:\n- Area of regular hexagon\n- Calculate area of inscribed hexagon\n- Regular hexagon perimeter and area\n- Formula for area of regular hexagon\n- Geometry problem solution: hexagon in circle\n- Hexagon inscribed in circle calculation", "Optimize your content with these terms to improve search visibility for geometry learners and math-related queries."]
