1. **Problem statement:** An equilateral triangle is inscribed in a circle of radius 10 cm. Find the area of the triangle. 2. **Formula and important rules:** - The radius $R$ of the circumscribed circle of an equilateral triangle with side length $a$ is given by: $$R = \frac{a}{\sqrt{3}}$$ - Rearranging to find $a$: $$a = R \sqrt{3}$$ - The area $A$ of an equilateral triangle with side length $a$ is: $$A = \frac{\sqrt{3}}{4} a^2$$ 3. **Calculate the side length $a$:** $$a = 10 \times \sqrt{3} = 10\sqrt{3}$$ 4. **Calculate the area $A$:** $$A = \frac{\sqrt{3}}{4} (10\sqrt{3})^2$$ 5. **Simplify the expression:** $$A = \frac{\sqrt{3}}{4} \times 100 \times 3 = \frac{\sqrt{3}}{4} \times 300$$ 6. **Multiply:** $$A = 75 \sqrt{3}$$ 7. **Final answer:** The area of the equilateral triangle inscribed in a circle of radius 10 cm is: $$\boxed{75 \sqrt{3} \text{ cm}^2}$$