Triangle Calculator
Solve any triangle using SSS, SAS, ASA/AAS, or SSA methods
How to Use This Triangle Calculator
- Choose which information you have: SSS (3 sides), SAS (2 sides + included angle), ASA/AAS (2 angles + 1 side), or SSA (2 sides + non-included angle)
- Enter the known values in the selected tab
- Click 'Solve Triangle' to find all missing sides and angles
- Review the area, perimeter, and triangle classification
Example: Given sides a=5, b=6, c=7 (SSS), the calculator finds angles A=44.42 degrees, B=57.12 degrees, C=78.46 degrees. Area is 14.70 square units. This is an acute scalene triangle since all angles are under 90 degrees and all sides are different lengths.
Tip: SSA (two sides and a non-included angle) can produce zero, one, or two valid triangles. The calculator shows all valid solutions when this 'ambiguous case' occurs.
Why Use a Triangle Calculator?
Solving triangles is fundamental to surveying, navigation, engineering, and construction. When you can't directly measure something, triangles often provide the solution.
- Survey land by measuring angles and one baseline distance
- Calculate roof dimensions from pitch angle and wall measurements
- Navigate using triangulation from known landmarks
- Design truss structures with specific angles and spans
- Calculate distances across rivers, canyons, or other obstacles
- Solve geometry homework and engineering problems
Understanding Your Results
The calculator returns all six triangle measurements plus area and perimeter. Classification tells you the triangle type.
| Result | Meaning | Action |
|---|---|---|
| One angle = 90 degrees | Right triangle | Pythagorean theorem applies: a² + b² = c² |
| All angles < 90 degrees | Acute triangle | All altitudes fall inside the triangle |
| One angle > 90 degrees | Obtuse triangle | One altitude falls outside the triangle |
| Two or more sides equal | Isosceles or equilateral | Equal sides have equal opposite angles |
Meaning: Right triangle
Action: Pythagorean theorem applies: a² + b² = c²
Meaning: Acute triangle
Action: All altitudes fall inside the triangle
Meaning: Obtuse triangle
Action: One altitude falls outside the triangle
Meaning: Isosceles or equilateral
Action: Equal sides have equal opposite angles
Note: Triangle inequality: any side must be less than the sum of the other two sides. If violated, no valid triangle exists.
About Triangle Calculator
Formula
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) | Law of Cosines: c² = a² + b² - 2ab·cos(C) Law of Sines works when you know an angle and its opposite side. Law of Cosines works when you know all three sides or two sides with their included angle.
Current Standards: In surveying, angles are typically measured to the nearest arc-second (1/3600 of a degree) for precision work. Engineering drawings usually specify angles to one decimal place.
Frequently Asked Questions
What is the 'ambiguous case' in SSA problems?
When you know two sides and an angle opposite one of them (not between them), multiple triangles may satisfy the conditions. The known angle can pair with two different triangles having that angle acute or obtuse. The calculator shows both solutions when they exist. If the side opposite the known angle is too short to reach the other side, no triangle exists.
How do I know which formula method to use?
Use SSS when you have all three sides. Use SAS when you have two sides and the angle between them. Use ASA or AAS when you have two angles (the third is automatic) and any side. Use SSA when you have two sides and an angle not between them - but watch for the ambiguous case.
What's the difference between altitude and slant height?
Altitude (height) is the perpendicular distance from a vertex to the opposite side (or its extension). It forms a 90-degree angle with the base. Slant height typically refers to pyramids and cones - the distance along the sloped face. In triangles, we use altitude for area calculations: Area = (1/2) × base × height.
Why do angles in a triangle always sum to 180 degrees?
This is a fundamental property of Euclidean (flat) geometry. Imagine walking around a triangle: at each corner, you turn by the exterior angle. After three turns, you've made one complete rotation (360°). Since each exterior angle is 180° minus the interior angle, the interior angles must sum to 180°. On curved surfaces (like a sphere), this rule doesn't hold.
How accurate are the calculations?
The calculator uses standard trigonometric functions accurate to about 15 significant figures. Results are displayed to 4 decimal places, which is more precision than most practical measurements require. The main source of error is usually the input measurements, not the calculation itself.