Triangle Angle Calculator: Solve Angles & Sides Online
A triangle angle calculator solves for missing interior angles, side lengths, perimeter, and area of any triangle given three known dimensions using the Law of Sines and Law of Cosines. Enter your known sides or angles using SSS, SAS, ASA, AAS, or Right Triangle rules to calculate all geometric parameters instantly.
Triangle Solution Readout
RIGHT TRIANGLEHow to Use This Tool
- Choose your triangle solving mode from the dropdown: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), AAS, or Right Triangle.
- Enter your known positive numerical values into the parameter inputs.
- View the instant solution including all three interior angles, side lengths, perimeter, area, and geometric classification.
- Review the step-by-step trigonometric formulas applied for your configuration below.
- Click 'Reset' to clear all fields and try another triangle.
What the Triangle Angle Calculator Does
The Triangle Angle Calculator is a trigonometric solver that computes all unknown dimensions of a planar Euclidean triangle from any valid combination of three known parameters. Supporting SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and Right-Triangle configurations, the instrument outputs all three interior angles in degrees, all three side lengths, total perimeter, enclosed area, and geometric classification.
In surveying, navigation, structural framing, and CNC fabrication, triangles form the fundamental building blocks of rigid structures. If a single angle or diagonal brace length is unknown, this calculator resolves the missing metrics using proven trigonometric theorems.
Mathematical Formulas & Triangle Theorems
The solver applies four universal mathematical principles depending on the selected solving mode:
1. Law of Cosines
c² = a² + b² - 2ab · cos(γ)
cos(γ) = (a² + b² - c²) / (2ab)
Generalizes the Pythagorean theorem to any oblique triangle; solves SSS and SAS.
2. Law of Sines
a / sin(α) = b / sin(β) = c / sin(γ)
Relates opposing sides and angles; solves ASA and AAS configurations.
3. Triangle Angle Sum Rule
α + β + γ = 180.00°
In Euclidean planar geometry, the sum of all three interior angles is always identically 180°.
4. Heron's Area Formula
s = (a + b + c) / 2, Area = √[s · (s - a) · (s - b) · (s - c)]
Computes enclosed area directly from side lengths without requiring an altitude measurement.
Two Worked Practical Examples
Example 1: Solving SSS for a 5-7-10 Structural Truss
Known: Side lengths a = 5 m, b = 7 m, c = 10 m.
- Verify Triangle Inequality: 5 + 7 = 12 > 10. The triangle is physically valid.
- Apply Law of Cosines to solve Angle γ (opposite c = 10):
cos(γ) = (5² + 7² - 10²) / (2 × 5 × 7) = (25 + 49 - 100) / 70 = -26 / 70 ≈ -0.3714
γ = arccos(-0.3714) ≈ 111.80° (Obtuse Angle) - Solve Angle α (opposite a = 5):
cos(α) = (7² + 10² - 5²) / (2 × 7 × 10) = (49 + 100 - 25) / 140 = 124 / 140 ≈ 0.8857 → α ≈ 27.66° - Solve remaining angle: β = 180° - 111.80° - 27.66° = 60.54°.
- Calculate Area via Heron's: s = (5 + 7 + 10) / 2 = 11, Area = √(11 × 6 × 4 × 1) = √264 ≈ 16.25 m².
Example 2: Solving SAS for Property Boundary Survey
Known: Two fence lines a = 50 ft and b = 70 ft meet at corner angle γ = 60.00°.
- Calculate missing boundary side c:
c² = 50² + 70² - 2(50)(70)·cos(60°) = 2,500 + 4,900 - 7,000(0.5) = 7,400 - 3,500 = 3,900
c = √3,900 ≈ 62.45 ft - Apply Law of Sines to find α:
sin(α) = (a · sin(γ)) / c = (50 × sin(60°)) / 62.45 = (50 × 0.8660) / 62.45 ≈ 0.6934 → α ≈ 43.90° - Remaining angle: β = 180° - 60° - 43.90° = 76.10°.
- Enclosed Area: Area = 0.5 × 50 × 70 × sin(60°) = 1,750 × 0.8660 ≈ 1,515.5 sq ft.
Special Triangles & Pythagorean Triples Reference Table
These classic geometric triangles appear constantly across engineering, carpentry framing, and trigonometry:
| Triangle Name | Side Ratios (a : b : c) | Interior Angles (α - β - γ) | Area Factor | Classification |
|---|---|---|---|---|
| 3-4-5 Right Triangle | 3 : 4 : 5 | 36.87° - 53.13° - 90.00° | 6.00 | Integer Right |
| 5-12-13 Right Triangle | 5 : 12 : 13 | 22.62° - 67.38° - 90.00° | 30.00 | Integer Right |
| 8-15-17 Right Triangle | 8 : 15 : 17 | 28.07° - 61.93° - 90.00° | 60.00 | Integer Right |
| 7-24-25 Right Triangle | 7 : 24 : 25 | 16.26° - 73.74° - 90.00° | 84.00 | Integer Right |
| 30-60-90 Special Right | 1 : √3 (~1.732) : 2 | 30.00° - 60.00° - 90.00° | √3/2 (~0.866) | Special Right |
| 45-45-90 Isosceles Right | 1 : 1 : √2 (~1.414) | 45.00° - 45.00° - 90.00° | 0.500 | Special Right |
| Equilateral Triangle | 1 : 1 : 1 | 60.00° - 60.00° - 60.00° | √3/4 (~0.433) | Equilateral |
Common Mistakes When Solving Triangles
- Violating the Triangle Inequality: Attempting to solve SSS with side lengths like 2, 3, 6. Because 2 + 3 = 5 < 6, the shorter sides cannot meet, making a closed triangle geometrically impossible.
- Angle Sum Greater Than 180°: Entering two angles that sum to 180° or more in ASA/AAS mode leaves zero degrees for the third interior angle.
- Mismatched Radian/Degree Modes: Trigonometric formulas require angle arguments in radians; our solver automatically handles degree conversion to avoid precision errors.
- The Ambiguous SSA Case: Providing two sides and a non-included angle can yield two distinct valid triangles, one triangle, or zero triangles. Our solver prioritizes mathematically unambiguous configurations (SSS, SAS, ASA, AAS).
Frequently Asked Questions: Triangle Calculator
Mathematical explanations on solving oblique and right triangles using the Laws of Sines and Cosines.
What can this Triangle Calculator solve?
This Triangle Calculator solves for all unknown interior angles (α, β, γ), side lengths (a, b, c), perimeter, and surface area of any triangle. It supports five primary geometric solving configurations: SSS (three known sides), SAS (two sides and their included angle), ASA (two angles and their included side), AAS (two angles and a non-included side), and right triangles, applying the Pythagorean theorem, Law of Sines, and Law of Cosines.
What is the minimum information needed to solve a triangle?
To solve a general oblique triangle, you must know at least three independent geometric parameters, with at least one parameter being a side length. Knowing three angles alone (AAA) determines triangle shape and proportions but cannot establish scale or absolute side lengths. For right triangles, knowing any two side lengths, or one side length and one acute angle, is sufficient because the 90° right angle provides the third constraint.
Which values can this right triangle calculator find?
Given any two known sides—such as both legs (a and b), or one leg and the hypotenuse (c)—this right triangle calculator finds the missing side length using the Pythagorean theorem (a² + b² = c²), both acute interior angles (α and β) via inverse trigonometric functions (arcsin, arccos, arctan), the total perimeter (a + b + c), and the exact surface area (0.5 × a × b).
How We Calculate
Triangles are solved using exact trigonometric identities. SSS and SAS configurations apply the Law of Cosines cos(C) = (a² + b² - c²) / (2ab), while ASA and AAS apply the Law of Sines a / sin(A) = b / sin(B). All inputs undergo automated triangle inequality verification before evaluation.
Last updated: September 2026
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