Need to solve a triangle? Our free Triangle Calculator handles all triangle types and solving cases. Whether you have three sides (SSS), two sides and an angle (SAS, SSA), or two angles and a side (ASA, AAS), this tool calculates all unknown values instantly. Find missing sides and angles, calculate area using multiple methods, determine perimeter, heights, and special points like centroid and circumcenter. Perfect for students studying geometry, engineers working on structural designs, architects calculating roof angles, or anyone who needs accurate triangle calculations. Supports both degrees and radians, handles the ambiguous SSA case, and validates triangle existence. All calculations happen instantly in your browser with step-by-step formulas. No signups, no limits, completely free.
A triangle calculator is a mathematical tool that determines all unknown properties of a triangle when given sufficient known information. A triangle is defined by six elements: three sides (typically labeled a, b, c) and three angles (labeled A, B, C, opposite their respective sides). Given any three independent pieces of information (with at least one side), a triangle can be uniquely determined (except for the ambiguous SSA case). Our calculator supports all standard triangle solving cases: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), AAS (two angles and non-included side), SSA (two sides and non-included angle - ambiguous case), and right triangles (one 90° angle). The calculator applies appropriate geometric laws including the Law of Sines, Law of Cosines, and Pythagorean theorem to find missing values. It also computes derived properties like area (using Heron's formula, trigonometric formulas, or base-height methods), perimeter, semi-perimeter, heights to each side, inradius, circumradius, and classifies the triangle by angles (acute, obtuse, right) and sides (equilateral, isosceles, scalene).
Our calculator supports all triangle solving cases (SSS, SAS, ASA, AAS, SSA, right triangle), calculates area using multiple methods including Heron's formula, finds all missing sides and angles automatically, computes perimeter, heights, inradius, and circumradius, handles the ambiguous SSA case with multiple solutions, classifies triangles by angles and sides, supports both degree and radian angle input, and validates triangle existence using the triangle inequality theorem.
Using the Triangle Calculator is straightforward: First, select your known values. Choose from the available cases: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), SSA (two sides + non-included angle), or right triangle (one 90° angle + other info). Next, enter your known measurements in the input fields. Provide the side lengths and/or angle measures you have. Select your preferred units (degrees or radians for angles). The calculator validates your inputs for triangle inequality (for SSS) and valid angle ranges. Then click Calculate to solve. The calculator determines which geometric laws to apply based on your inputs: For SSS: Uses Law of Cosines to find angles, Heron's formula for area. For SAS: Uses Law of Cosines to find third side, then other methods for remaining values. For ASA/AAS: Uses angle sum property (180°) to find third angle, then Law of Sines for sides. For SSA: Uses Law of Sines, checks for ambiguous case (0, 1, or 2 solutions). For right triangles: Uses Pythagorean theorem and basic trig ratios. Finally, review the complete solution. The calculator displays all six elements (three sides, three angles), area (calculated multiple ways for verification), perimeter and semi-perimeter, heights/altitudes to each side, inradius and circumradius, triangle classification, and special points (centroid, orthocenter, etc.).
Geometry students solve triangle problems involving angles, sides, and area calculations. Architects calculate roof pitches and structural angles. Surveyors determine property boundaries and land measurements. Engineers analyze force triangles and vector components. Physics students work with vector addition and resolution. Carpenters ensure corners are square and cuts are accurate. Navigation applications calculate distances and bearings between points. Game developers implement collision detection and geometry in graphics.
Complete Triangle Solutions - Get all unknown values, not just one or two. Our calculator provides a complete geometric analysis including sides, angles, area, perimeter, heights, and special points. Handles All Cases - Whether you have SSS, SAS, ASA, AAS, SSA, or a right triangle, the calculator automatically applies the correct geometric laws. Even handles the ambiguous SSA case with multiple solutions. Multiple Area Methods - Area is calculated using different methods based on available data (Heron's formula, trigonometric formulas, base-height), providing cross-verification of accuracy. Input Validation - The calculator checks triangle inequality for sides and valid ranges for angles, alerting you immediately if the inputs cannot form a valid triangle. Educational Value - See which formulas are applied and understand the geometric principles. Perfect for students learning trigonometry and geometry. Step-by-Step Understanding - While results are instant, the logical flow follows standard geometric principles, helping users understand the relationships between triangle elements. Professional Accuracy - Engineers, architects, and designers need precise calculations. Our calculator uses double-precision arithmetic and standard mathematical formulas for reliable results. Free & Accessible - No expensive graphing calculator or specialized software needed. Access professional triangle solving from any browser, completely free with no limits.
Students - High school and college students studying geometry, trigonometry, and precalculus use this tool for homework, exam preparation, and concept verification. See how the Law of Sines and Law of Cosines work in practice. Teachers & Educators - Create lesson materials, generate practice problems, and demonstrate triangle-solving concepts. Verify student work and illustrate geometric principles with accurate examples. Engineers - Civil, mechanical, and structural engineers regularly solve triangles for truss analysis, structural load calculations, mechanism design, and surveying applications. Architects - Calculate roof pitches, stair angles, bracing requirements, and spatial layouts. Triangle geometry is fundamental to architectural design and drafting. Surveyors & Cartographers - Triangulation is the basis of land surveying and map-making. Calculate distances, angles, and positions from known reference points. Computer Graphics Developers - 3D modeling, game development, and rendering all rely heavily on triangle calculations. Understand the geometry behind mesh operations and transformations. Physics Students & Researchers - Vector analysis, force diagrams, optics problems, and wave interference all involve triangular relationships. Astronomers use parallax (triangular method) to measure stellar distances. DIY Enthusiasts & Makers - Calculate angles and lengths for woodworking projects, home renovations, garden layouts, and craft projects involving triangular designs. Test Prep Students - SAT, ACT, GRE, and other standardized tests include triangle problems. Practice with a tool that reinforces the underlying concepts.
Select what you know about your triangle: three sides (SSS), two sides and included angle (SAS), two angles and side (ASA), or other combination. Enter the known values in the appropriate fields—angles in degrees, sides as lengths. Click Calculate to solve for unknown angles, sides, and area. Review the step-by-step solution showing which formulas were applied (Law of Cosines, Law of Sines, or Heron's formula). The calculator determines triangle validity and checks for impossible combinations. View the triangle visualization if available. All results are computed together for consistency.
Always verify your triangle is valid: sum of any two sides must exceed the third side. Check that angles sum to exactly 180° for valid triangles. Use the most precise input values available to get accurate results. For ambiguous cases (SSA), the calculator may show two possible solutions. Round final answers appropriately for your application—typically match input precision. Verify your calculated values by working backwards through the formulas.
This calculator solves standard Euclidean triangles in 2D space. It does not handle spherical triangles (on curved surfaces like Earth navigation). Very flat or extreme triangles may have numerical precision issues. Degenerate triangles (zero area) are identified but may have limited solution information. Complex triangle problems requiring iterative solving may need specialized software. The calculator assumes flat geometry, not curved spacetime or relativistic scenarios.
A triangle calculator is a mathematical tool that solves for unknown values of a triangle when given sufficient known values. A triangle has 6 fundamental elements: 3 sides (a, b, c) and 3 angles (A, B, C). Our calculator can solve any triangle given various combinations of these elements: SSS (Side-Side-Side) - All three sides known. Calculates all angles and area using the Law of Cosines. SAS (Side-Angle-Side) - Two sides and the included angle known. Most straightforward case; calculates the third side using Law of Cosines. ASA (Angle-Side-Angle) - Two angles and the included side known. Calculates remaining angle (180° - A - B) and other sides using Law of Sines. AAS (Angle-Angle-Side) - Two angles and a non-included side known. Similar to ASA; first find the third angle. SSA (Side-Side-Angle) - Two sides and a non-included angle known. The ambiguous case; may have 0, 1, or 2 valid solutions. Right Triangle - One 90° angle. Can use simple Pythagorean theorem and basic trigonometry. The calculator provides: All missing sides and angles, Area using multiple methods (Heron's formula, base×height/2, trigonometric formulas), Perimeter, Heights/altitudes to each side, Inradius and circumradius, Classification (acute, obtuse, right, equilateral, isosceles, scalene).
There are several methods to calculate triangle area depending on what values you know: Base and Height: Area = (base × height) / 2. The most fundamental formula. Requires knowing one side and its corresponding altitude. Heron's Formula: When all three sides (a, b, c) are known. First calculate semi-perimeter: s = (a+b+c)/2. Then Area = √[s(s-a)(s-b)(s-c)]. No angles needed. Two Sides and Included Angle (SAS): Area = (1/2) × a × b × sin(C). Useful when you know two sides and the angle between them. Two Angles and a Side: Area = (a² × sin(B) × sin(C)) / (2 × sin(A)). Useful for ASA or AAS cases. Using Coordinates: If vertices are at (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. Inradius Method: Area = r × s, where r is the inradius and s is the semi-perimeter. Circumradius Method: Area = (a × b × c) / (4 × R), where R is the circumradius. Our calculator automatically selects and applies the most appropriate formula based on your inputs, and often calculates area using multiple methods to verify accuracy.
Triangle congruence cases determine when two triangles are identical (congruent) and how many triangles can be formed given certain information: SSS (Side-Side-Side): Given three sides. Always determines a unique triangle (if triangle inequality is satisfied). Use Law of Cosines to find angles. SAS (Side-Angle-Side): Given two sides and the included angle. Always determines a unique triangle. Use Law of Cosines to find the third side. ASA (Angle-Side-Angle): Given two angles and the included side. Always determines a unique triangle. Find third angle (180° - sum), then use Law of Sines. AAS (Angle-Angle-Side): Given two angles and a non-included side. Always determines a unique triangle. Find third angle first, then use Law of Sines. SSA (Side-Side-Angle) - The Ambiguous Case: Given two sides and a non-included angle. May have 0, 1, or 2 solutions! Depends on whether the given angle is acute or obtuse, and the relationship between sides. This is called the ambiguous case because it's not always clear how many triangles satisfy the conditions. Our calculator handles all these cases automatically, including detecting and reporting when the SSA case has multiple valid solutions.
Right triangles (with one 90° angle) are the simplest to solve and have special properties: The Pythagorean Theorem: a² + b² = c², where c is the hypotenuse (side opposite the right angle). If you know any two sides, you can find the third. Trigonometric Ratios: For angle A (not the right angle): sin(A) = opposite/hypotenuse = a/c, cos(A) = adjacent/hypotenuse = b/c, tan(A) = opposite/adjacent = a/b. Complementary Angles: The two non-right angles sum to 90°. If one acute angle is 30°, the other is 60°. Special Right Triangles: 45-45-90 Triangle: Sides are in ratio 1:1:√2. If legs are length 1, hypotenuse is √2. 30-60-90 Triangle: Sides are in ratio 1:√3:2. Short leg (opposite 30°) is half the hypotenuse. Area of Right Triangle: Area = (1/2) × leg₁ × leg₂ (since legs are perpendicular, one serves as base, other as height). To solve a right triangle, you only need to know: Any two sides, or One side and one non-right angle. Our right triangle calculator automatically applies these formulas and can also calculate additional values like altitude to hypotenuse, inradius, and circumradius (which equals half the hypotenuse for right triangles).
The Law of Sines and Law of Cosines are fundamental tools for solving any triangle: Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R (where R is the circumradius). This law states that the ratio of any side to the sine of its opposite angle is constant for all three sides. Best used when you know: AAS or ASA cases (two angles and a side), or SSA case (ambiguous case). You can set up proportions like: a/sin(A) = b/sin(B) to find missing values. Law of Cosines: c² = a² + b² - 2ab·cos(C). This generalizes the Pythagorean theorem for any triangle. Variations: a² = b² + c² - 2bc·cos(A), b² = a² + c² - 2ac·cos(B). Best used when you know: SSS case (three sides) - find any angle, SAS case (two sides and included angle) - find the third side. For right triangles, the Law of Cosines reduces to the Pythagorean theorem (cos(90°) = 0). When to Use Each: Use Law of Cosines first when you have SSS or SAS, as it directly gives you the missing element. Use Law of Sines when you have angle-side-angle relationships. Our calculator automatically selects the appropriate law based on your inputs and applies it correctly.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is a fundamental requirement for three lengths to form a valid triangle. The Three Conditions: a + b > c, a + c > b, b + c > a. If any of these conditions is not met, the three lengths cannot form a triangle. Equivalent Form: The difference of any two sides must be less than the third side: |a - b| < c, |a - c| < b, |b - c| < a. Why This Matters: Validation - Before solving, always check if given sides can form a triangle. SSA Ambiguity - In the SSA case, the triangle inequality helps determine if 0, 1, or 2 solutions exist. Degenerate Triangle - If a + b = c exactly, the 'triangle' collapses into a line segment (degenerate triangle with zero area). Practical Example: Can sides 3, 4, 10 form a triangle? Check: 3 + 4 = 7, which is NOT > 10. Therefore, these sides cannot form a triangle. Can sides 5, 7, 9 form a triangle? Check: 5+7=12>9 ✓, 5+9=14>7 ✓, 7+9=16>5 ✓. Yes, valid triangle. Our calculator automatically validates inputs against the triangle inequality and alerts you if the given values cannot form a valid triangle.
Triangles have several important lines, points, and associated circles: Altitudes (Heights): Perpendicular lines from each vertex to the opposite side (or its extension). The three altitudes intersect at the orthocenter. Medians: Lines from each vertex to the midpoint of the opposite side. The three medians intersect at the centroid, which divides each median in a 2:1 ratio. The centroid is the triangle's center of mass. Angle Bisectors: Lines that bisect each angle. The three angle bisectors intersect at the incenter, which is the center of the incircle (inscribed circle). The inradius (r) is the radius of the incircle. Perpendicular Bisectors: Lines perpendicular to each side at its midpoint. The three perpendicular bisectors intersect at the circumcenter, which is the center of the circumcircle (circumscribed circle). The circumradius (R) is the radius of this circle. For right triangles, the circumcenter is at the midpoint of the hypotenuse. Important Formulas: Inradius: r = Area / s (where s is semi-perimeter), Circumradius: R = (abc) / (4 × Area), Distance between circumcenter and orthocenter (Euler line): OH² = 9R² - (a²+b²+c²). Our calculator computes these special points and associated values when solving a triangle, providing a complete geometric analysis.
Triangles are fundamental shapes with countless practical applications: Architecture & Construction: Roof trusses use triangular frameworks for structural stability. Bridges employ triangular support structures. Staircase calculations use right triangle geometry. Surveying & Navigation: Triangulation determines distances to distant points. GPS systems use triangular positioning from satellites. Map-making relies on triangular networks of known points. Engineering: Truss bridges and towers use triangular units for strength. Mechanism design uses linkages that form triangles. Stress analysis of structures involves triangular elements. Computer Graphics: 3D models are built from triangular meshes. Rendering uses triangle rasterization. Texture mapping applies 2D images to 3D triangular surfaces. Physics & Astronomy: Triangulation measures distances to stars and planets. Parallax calculations use triangular geometry. Optics problems often involve triangular paths. Sports & Recreation: Billiards uses angle calculations (triangular paths). Rock climbing route planning involves angle assessments. Sailboat navigation uses triangular course calculations. Everyday Life: Ladder safety (right triangle with ground and wall). Finding the shortest path (triangle inequality). Photography composition (rule of thirds relates to triangles). The study of triangles (trigonometry) is essential for careers in engineering, architecture, physics, computer science, and many technical fields. Our calculator supports all these applications by providing accurate triangle solutions for any valid input.