Triangle Calculator

## Complete Triangle Calculator: Solve Any Triangle This comprehensive triangle calculator solves triangles using any valid combination of sides and angles. Whether you know three sides (SSS), two sides and an angle (SAS), or two angles and a side (ASA/AAS), this tool calculates all remaining measurements plus area, perimeter, altitudes, medians, and more. ### Understanding Triangle Calculation Methods **SSS (Side-Side-Side)**: When you know all three side lengths, the triangle is uniquely determined. The calculator uses the Law of Cosines to find all angles. **SAS (Side-Angle-Side)**: When you know two sides and the angle between them (included angle), the Law of Cosines finds the third side, then the Law of Sines finds the remaining angles. **ASA (Angle-Side-Angle)**: When you know two angles and the side between them, the third angle is 180° minus the other two. The Law of Sines then finds the remaining sides. **AAS (Angle-Angle-Side)**: Similar to ASA, but the known side is not between the known angles. The same laws apply to solve the triangle. ### The Law of Cosines The Law of Cosines generalizes the Pythagorean theorem to all triangles: **c² = a² + b² - 2ab·cos(C)** This formula relates any side to the other two sides and their included angle. It's essential for SSS and SAS calculations. When angle C equals 90°, the formula reduces to the Pythagorean theorem since cos(90°) = 0. ### The Law of Sines The Law of Sines establishes a proportion between sides and their opposite angles: **a/sin(A) = b/sin(B) = c/sin(C) = 2R** Where R is the circumradius. This law is fundamental for ASA, AAS, and finding missing angles in SAS problems. ### Triangle Types by Sides **Equilateral**: All three sides are equal (a = b = c). All angles are 60°. This is the most symmetric triangle. **Isosceles**: Two sides are equal. The angles opposite equal sides are also equal. Many geometric proofs use isosceles triangles. **Scalene**: All three sides have different lengths. All three angles are different. This is the most general triangle type. ### Triangle Types by Angles **Acute Triangle**: All angles are less than 90°. For sides a ≤ b ≤ c, we have a² + b² > c². **Right Triangle**: One angle equals exactly 90°. The Pythagorean theorem applies: a² + b² = c². **Obtuse Triangle**: One angle exceeds 90°. For the largest side c, we have a² + b² < c². ### Area Formulas **Base and Height**: Area = ½ × base × height. The most intuitive formula, requiring perpendicular height. **Heron's Formula**: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter. Works with just the three sides. **Two Sides and Included Angle**: Area = ½ × a × b × sin(C). Useful when you know SAS. **Using Circumradius**: Area = abc/(4R). Relates area to sides and circumradius. **Using Inradius**: Area = r × s, where r is inradius and s is semi-perimeter. ### Special Points and Circles **Centroid**: The intersection of medians. Located at the "center of mass" of the triangle, dividing each median in a 2:1 ratio. **Circumcenter**: The center of the circumscribed circle (circumcircle). Equidistant from all three vertices. **Incenter**: The center of the inscribed circle (incircle). Equidistant from all three sides. **Orthocenter**: The intersection of altitudes. Location depends on triangle type (inside for acute, on vertex for right, outside for obtuse). ### Altitudes, Medians, and Perpendicular Bisectors **Altitudes**: Perpendicular segments from each vertex to the opposite side. Height formulas: h_a = 2·Area/a. **Medians**: Segments from each vertex to the midpoint of the opposite side. Formula: m_a = ½√(2b² + 2c² - a²). **Perpendicular Bisectors**: Lines through side midpoints perpendicular to that side. They meet at the circumcenter. ### Real-World Applications **Architecture and Construction**: Triangles provide structural stability. Roof trusses, bridges, and building frames rely on triangular geometry. **Navigation and Surveying**: Triangulation determines positions by measuring angles to known points. GPS uses similar principles in 3D. **Engineering**: Stress analysis, force decomposition, and structural design all use triangle calculations. **Computer Graphics**: 3D models consist of triangular meshes. Triangle calculations are fundamental to rendering. **Art and Design**: Triangular compositions create visual interest. The golden triangle appears in classical art. ### The Triangle Inequality For any valid triangle with sides a, b, c: - a + b > c - a + c > b - b + c > a If any inequality fails, the sides cannot form a triangle. This calculator validates inputs automatically. ### Common Angle Relationships **Sum of Angles**: A + B + C = 180° for any triangle. **Exterior Angles**: An exterior angle equals the sum of the two non-adjacent interior angles. **Largest Side vs. Angle**: The largest side is always opposite the largest angle. ### Special Triangles **30-60-90 Triangle**: Sides in ratio 1 : √3 : 2. Half of an equilateral triangle. **45-45-90 Triangle**: Sides in ratio 1 : 1 : √2. Half of a square. **3-4-5 Triangle**: The simplest right triangle with integer sides. **Golden Triangle**: Isosceles triangle with sides in golden ratio (1.618...). ### Using This Calculator Enter your known values based on the selected method: - **SSS**: All three side lengths - **SAS**: Two sides and the angle between them - **ASA**: Two angles and the side between them - **AAS**: Two angles and any side The calculator instantly provides all remaining values, including area, perimeter, all altitudes, all medians, inradius, circumradius, and classifications. Whether you're solving geometry homework, designing structures, or analyzing spatial relationships, this triangle calculator provides complete solutions for any valid triangle configuration.