1. **Problem:** Given points $A=(-3,-1)$ and $B=(2,5)$ are two vertices of an equilateral triangle $ABC$. Find the area of this triangle. 2. **Formula and rules:** - The length of side $AB$ is calculated by the distance formula: $$AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$$ - In an equilateral triangle, all sides are equal. - The height $h$ of an equilateral triangle with side length $a$ is: $$h = \frac{\sqrt{3}}{2} a$$ - The area $P$ of an equilateral triangle is: $$P = \frac{\sqrt{3}}{4} a^2$$ 3. **Calculate side length $AB$:** $$AB = \sqrt{(2 - (-3))^2 + (5 - (-1))^2} = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}$$ 4. **Calculate area:** $$P = \frac{\sqrt{3}}{4} (\sqrt{61})^2 = \frac{\sqrt{3}}{4} \times 61 = \frac{61 \sqrt{3}}{4}$$ 5. **Answer:** The area of the equilateral triangle $ABC$ is $$\boxed{\frac{61 \sqrt{3}}{4}}$$