Law of Sines Calculator — Law of Cosines, Pythagorean Theorem & More

This math calculator covers six tools: law of sines, law of cosines, Pythagorean theorem, right triangle calculator, volume of a cylinder, and interval of convergence. Law of Sines: Enter three known values (sides and angles) and leave one blank. The law of sines calculator solves for the missing side or angle using a/sin(A) = b/sin(B). Law of Cosines: Enter two sides and the included angle (SAS) to find the third side, or three sides (SSS) to find an angle. Formula: c² = a² + b² − 2ab·cos(C). Pythagorean Theorem: Enter any two sides of a right triangle. The calculator finds the missing side, area, and perimeter. Right Triangle Calculator: Enter any combination of two sides or one side and one acute angle. The calculator finds all remaining sides and angles. Volume of a Cylinder: Enter the radius and height. The calculator returns the volume (πr²h), lateral surface area (2πrh), and total surface area (2πr² + 2πrh). Interval of Convergence: Enter the ratio test limit L and the series center c. Returns the radius R = 1/L and the convergence interval.

**Law of Sines Calculator — Complete Guide** The law of sines applies to any triangle. Formula: a/sin(A) = b/sin(B) = c/sin(C). Use it for AAS (two angles + one side), ASA (two angles + included side), and SSA cases. The SSA case is the "ambiguous case" — it can yield 0, 1, or 2 valid triangles. The law of sines solver above returns the principal (acute) solution. **Law of Cosines Calculator — Complete Guide** The law of cosines handles SAS and SSS triangles. For SAS (two sides, included angle): c² = a² + b² − 2ab·cos(C). For SSS (three sides, find angle): cos(C) = (a² + b² − c²) / (2ab). When the included angle is 90°, the formula reduces to the Pythagorean theorem. Use the law of cosines solver when the law of sines doesn't apply. **Pythagorean Theorem Calculator** For right triangles only: a² + b² = c² where c is the hypotenuse (longest side, opposite the 90° angle). This is one of the most fundamental formulas in mathematics and the foundation of trigonometry. To find the hypotenuse: c = √(a² + b²). To find a leg: a = √(c² − b²). The right triangle side calculator above handles all three cases. **Right Angle Triangle Calculator** A right triangle has one 90° angle. Knowing any two of the five values (a, b, c, angle A, angle B) is enough to find all others. The trigonometric relationships are: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b. This right triangle angle calculator uses these relationships for any combination of known values. **Volume of a Cylinder Calculator** A cylinder has a circular base with radius r and height h. Volume = πr²h. Total surface area = 2πr² + 2πrh. The factor 2πr² covers both circular bases; 2πrh covers the lateral (curved) surface. This formula appears in AP Physics, Calculus (solids of revolution), and SAT Math. **Interval of Convergence Calculator — Power Series** A power series Σ aₙ(x−c)ⁿ converges for all x within distance R of the center c, where R = 1/L and L is the ratio test limit. The interval of convergence is the open interval (c−R, c+R). Endpoints require separate testing: substitute x = c±R and check for convergence or divergence using the p-series test, alternating series test, or comparison test.