Solve any triangle — find sides, angles, area, perimeter, and height. Supports SSS, SAS, ASA, AAS, right triangles, and equilateral triangles with step-by-step solutions.
Enter any two values of a right triangle (legs a, b or hypotenuse c) and solve for everything else. Angle C is always 90°.
Calculate triangle area from different combinations of known values — no need to know all sides or angles.
A triangle has six measurements: three sides (a, b, c) and three angles (A, B, C), where each angle is opposite its corresponding side. Knowing any three of these — provided at least one is a side — is sufficient to determine the other three completely. This is what mathematicians call "solving the triangle."
SSS (Side-Side-Side): All three sides are known. Use the Law of Cosines to find any angle: cos(A) = (b² + c² − a²) ÷ 2bc. Once one angle is found, use the Law of Sines or repeat for others. This is the most common construction scenario — you measure all three sides on a job site and need the angles.
SAS (Side-Angle-Side): Two sides and the included angle (between them) are known. Use the Law of Cosines to find the third side, then the Law of Sines for remaining angles. Common in navigation and surveying when two legs of a path and the turning angle are measured.
ASA (Angle-Side-Angle): Two angles and the included side are known. The third angle = 180° − A − B. Then use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Used frequently in land surveying with theodolite measurements.
AAS (Angle-Angle-Side): Two angles and a non-included side. Same approach as ASA — find third angle first, then Law of Sines for remaining sides.
| Law | Formula | When to Use |
|---|---|---|
| Law of Sines | a/sin(A) = b/sin(B) = c/sin(C) | ASA, AAS, SSA (after checking ambiguity) |
| Law of Cosines (find side) | c² = a² + b² − 2ab·cos(C) | SAS — find the third side |
| Law of Cosines (find angle) | cos(A) = (b² + c² − a²) ÷ 2bc | SSS — find angles from all three sides |
| Pythagorean Theorem | c² = a² + b² | Right triangles only (special case of Law of Cosines) |
| Heron's Formula | Area = √(s(s−a)(s−b)(s−c)), s=(a+b+c)/2 | SSS — area from three sides |
| SAS Area | Area = ½·a·b·sin(C) | Two sides and included angle |
| Type | Definition | Properties |
|---|---|---|
| Acute | All angles < 90° | All altitudes inside triangle |
| Right | One angle = 90° | Pythagorean theorem applies; a² + b² = c² |
| Obtuse | One angle > 90° | Two altitudes fall outside triangle |
| Equilateral | All angles = 60°, all sides equal | Area = (√3/4)·a²; height = (√3/2)·a |
| Isosceles | Two equal sides and two equal angles | Base angles are equal; altitude from apex bisects base |
| Scalene | All sides different lengths | All angles different; most general case |
Triangles are the foundation of structural engineering. Unlike rectangles and other quadrilaterals, triangles are inherently rigid — the only polygon that cannot change shape without changing the length of its sides. This is why trusses, bridge designs, and roof structures rely entirely on triangulated members. Any rectangular frame can be stabilized by adding a single diagonal, which divides it into two triangles.
Roof pitch is expressed as rise over run — a 6/12 pitch means the roof rises 6 inches for every 12 inches of horizontal run. This forms a right triangle where the run is the base (leg b), the rise is the height (leg a), and the rafter length is the hypotenuse. For a 24-foot span with 6/12 pitch: half-span = 12 feet (run), rise = 12 × (6/12) = 6 feet, rafter = √(12² + 6²) = √(144 + 36) = √180 = 13.42 feet. Always add the eave overhang to the calculated rafter length.
Surveyors use triangulation to measure distances indirectly. A baseline of known length is established, then angles to a distant target point are measured from each end of the baseline. With two angles and the baseline (ASA case), the Law of Sines gives the distance to the target. This technique was used to map entire continents before GPS — the Great Trigonometric Survey of India (1802–1871) used triangulation to measure the height of Mount Everest from 150 miles away, getting within 26 feet of today's accepted value.
| Angles | Side Ratio | tan(A) | Common Use |
|---|---|---|---|
| 30° - 60° - 90° | 1 : √3 : 2 | tan(30°) = 0.577 | Half of equilateral triangle |
| 45° - 45° - 90° | 1 : 1 : √2 | tan(45°) = 1.000 | Square diagonal; isosceles right |
| 36° - 72° - 72° | 1 : φ : φ (golden ratio) | tan(36°) = 0.727 | Golden gnomon; pentagram |
| Equilateral 60-60-60 | 1 : 1 : 1 | tan(60°) = 1.732 | Strongest arch form |
The triangle inequality theorem states that the sum of any two sides must strictly exceed the third — this is not just a mathematical curiosity but a fundamental constraint that determines what shapes can exist in physical space. This has real implications: if you're building a triangular frame with two members of 6 feet and 8 feet, the connecting member must be strictly between 2 feet and 14 feet. At the extreme values, the triangle collapses to a straight line (degenerate triangle) with zero area. Engineers apply this principle constantly. In structural engineering, triangles are designed with generous proportions — highly obtuse triangles (one angle close to 180°) transmit forces poorly and are avoided in truss design.
The most common triangle area formula is Area = ½ × base × height, where the height must be perpendicular to the chosen base. For construction, the base is typically a measured horizontal distance and the height is the vertical rise. When height isn't directly measurable, Heron's Formula uses all three sides: with semi-perimeter s = (a+b+c)/2, Area = √(s·(s−a)·(s−b)·(s−c)). For SAS situations, Area = ½·a·b·sin(C) where C is the angle between sides a and b.
Triangles are fundamental in construction, navigation, and surveying. A carpenter uses the 3-4-5 right triangle to ensure corners are square — measure 3 feet along one wall, 4 feet along the adjacent wall, and if the diagonal is exactly 5 feet, the corner is a perfect 90°. A roofer uses triangle calculations to determine the roof pitch and rafter length from the span and rise. Land surveyors use triangulation — measuring angles from known baseline distances — to determine positions of distant points without directly measuring to them.
In navigation, the triangle formed by two bearings from different positions to the same landmark (called a "fix") is solved with the Law of Sines to find position. GPS systems still use this fundamental principle, just with satellites as the known points. Structural engineers use triangles because they are the only rigid polygon — adding a diagonal to any rectangular frame converts it into two triangles and prevents racking.
| Triangle Type | Angles | Side Ratios | Example (hyp = 10) |
|---|---|---|---|
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | a=5, b=8.66, c=10 |
| 45-45-90 (Isosceles right) | 45°, 45°, 90° | 1 : 1 : √2 | a=7.07, b=7.07, c=10 |
| 3-4-5 (Pythagorean triple) | 36.87°, 53.13°, 90° | 3 : 4 : 5 | a=6, b=8, c=10 |
| 5-12-13 | 22.62°, 67.38°, 90° | 5 : 12 : 13 | a=3.85, b=9.23, c=10 |
| 8-15-17 | 28.07°, 61.93°, 90° | 8 : 15 : 17 | a=4.71, b=8.82, c=10 |
| Equilateral | 60°, 60°, 60° | 1 : 1 : 1 | Area = (√3/4)·s² |