Triangle Calculator - Solve Any Triangle with Step-by-Step Workings
Every triangle problem reduces to one question: what do you already know, and what do you need to find? This calculator covers all five standard input combinations - SSS, SAS, ASA, AAS and SSA - and applies the correct formula automatically. You get not just the answer, but the full step-by-step working so you can understand and reproduce the solution, not just copy a number.
Which mode to choose: Know all 3 sides → SSS (Law of Cosines) · Know 2 sides + included angle → SAS (Law of Cosines) · Know 2 angles + included side → ASA (Law of Sines) · Know 2 angles + non-included side → AAS (Law of Sines) · Know 2 sides + non-included angle → SSA (ambiguous case) · Know only right angle + 2 values → Right Triangle tab
The Core Triangle Formulas - What Each One Does
📐 Area Formulas
- Base and height: Area = ½ × b × h. Simplest - but height must be perpendicular to the base
- Heron's formula (3 sides): Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2. No height needed
- SAS (2 sides + included angle): Area = ½ × a × b × sin(C)
- Right triangle: Area = ½ × leg₁ × leg₂
- Equilateral (side s): Area = (√3/4) × s²
🔺 Solving Formulas
- Law of Cosines: c² = a² + b² − 2ab·cos(C). Use for SSS and SAS
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Use for ASA, AAS, SSA
- Pythagorean theorem: a² + b² = c² (right triangles only)
- Angle sum: A + B + C = 180°. Always - find the third angle once two are known
- Inradius: r = Area ÷ s (s = semi-perimeter)
- Circumradius: R = (a × b × c) ÷ (4 × Area)
How to Solve Each Triangle Type - Step by Step
SSS (three sides known): Use the Law of Cosines to find any one angle first: cos(C) = (a² + b² − c²) / (2ab). Then use the Law of Sines for the second angle: sin(B)/b = sin(C)/c. Third angle from A + B + C = 180°. Area from Heron's formula.
SAS (two sides and included angle): Use Law of Cosines to find the third side: c² = a² + b² − 2ab·cos(C). Then use Law of Sines for the remaining angles. Area = ½ × a × b × sin(C).
ASA (two angles and included side): Find third angle: C = 180° − A − B. Use Law of Sines to find remaining sides: a = c × sin(A)/sin(C) and b = c × sin(B)/sin(C). Area from Heron's or base-height method.
AAS (two angles and non-included side): Same start as ASA - find third angle first. Then Law of Sines to find remaining two sides. Area from any method once all sides are known.
SSA - the ambiguous case: Two sides and a non-included angle can produce zero, one, or two valid triangles. Use Law of Sines to find sin(B): sin(B) = b × sin(A) / a. If sin(B) > 1: no triangle exists. If sin(B) = 1: one right triangle. If sin(B) < 1 and A is acute: potentially two triangles - B₁ = arcsin(sin(B)) and B₂ = 180° − B₁. Always check both solutions for validity.
Triangle Types - Classification by Sides and Angles
- Equilateral: All three sides equal. All three angles = 60°. Area = (√3/4)s². Most symmetric triangle type.
- Isosceles: Two sides equal. The angles opposite the equal sides are also equal. A common exam and real-world shape.
- Scalene: All three sides different lengths. All three angles different. The general triangle case.
- Right triangle: One angle exactly 90°. Satisfies the Pythagorean theorem: a² + b² = c². The hypotenuse is always the longest side, opposite the right angle.
- Obtuse triangle: One angle greater than 90°. The Law of Cosines gives a negative value for that angle's cosine. Only one obtuse angle is possible (angles must sum to 180°).
- Acute triangle: All three angles less than 90°. All cosines are positive. The most common triangle type in geometry problems.
The Inradius and Circumradius - What They Mean
Every triangle has two notable circles associated with it:
- Incircle (inradius r): The largest circle that fits entirely inside the triangle, touching all three sides. The centre (incentre) is equidistant from all three sides. Formula: r = Area ÷ s, where s is the semi-perimeter. For a 3-4-5 right triangle: s = 6, Area = 6, r = 6/6 = 1.
- Circumcircle (circumradius R): The circle that passes through all three vertices of the triangle. The centre (circumcentre) is equidistant from all three vertices. Formula: R = (a × b × c) ÷ (4 × Area). By the Law of Sines: a/sin(A) = 2R. For a 3-4-5 right triangle: R = (3×4×5)/(4×6) = 60/24 = 2.5 - which equals half the hypotenuse (correct for all right triangles).
A useful relationship: for any right triangle, the circumradius always equals exactly half the hypotenuse. This is because a right angle always inscribes a semicircle - the hypotenuse is a diameter of the circumcircle.
Where Triangle Geometry Appears in the Real World
- Construction and architecture: Roof trusses, bridge supports and structural frames are triangulated because triangles are the only polygon that is inherently rigid - no other shape maintains its form under load without additional internal supports
- Navigation and GPS: Triangulation uses known distances between reference points (satellites or towers) to determine an unknown position - the same Law of Sines and Cosines used here
- Surveying: Land boundaries and elevation maps are measured using triangulation from known baseline distances
- Physics: Force vectors are resolved into components using right triangle geometry. Any two forces combine into a resultant calculated by the parallelogram (triangle) law
- Computer graphics: All 3D objects in games and animation are built from triangulated meshes - every surface is broken into triangles because a triangle is always flat (three points define a unique plane)
- Astronomy: Stellar parallax uses the triangle formed by Earth's orbit and a nearby star to calculate the star's distance - the same principle as surveying but on an astronomical scale