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Triangle Calculator — SSS, SAS, ASA Trigonometry Studio

Solve triangles by entering any 3 values (SSS, SAS, ASA, AAS, SSA). Calculates missing angles, side lengths, area, perimeter, and inradius/circumradius. Free.

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Measurement Parameters

Side Dimensions

cm
cm
cm

Angle Measures (Degrees)

°
°
°
Quick Presets & Special Triangles

Vector Blueprint

Enter at least 3 dimensions to solve and view the dynamic blueprint.

Fundamental Triangle Rules

Master these core geometric principles used for all triangle calculations.

Angle Sum Property

The sum of all interior angles in any triangle always equals exactly 180 degrees.

∠A + ∠B + ∠C = 180°

Law of Sines

Used to find missing sides or angles when you have "opposite pairs" (Side/Angle matches).

a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines

Used to find a side when two sides and the included angle are known (SAS), or to find angles when all sides are known (SSS).

c² = a² + b² - 2ab·cos(C)

Triangle Inequality

For any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side.

a + b > c

Side-Angle Relationship

The longest side is always opposite the largest angle, and the shortest side is opposite the smallest angle.

Longest side ↔ Largest ∠

Exterior Angle Theorem

The measure of an exterior angle is equal to the sum of the two opposite interior angles. Total exterior angles sum to 360°.

Ext ∠A = ∠B + ∠C

Classification by Sides & Angles

Equilateral

All three sides are equal in length, and all interior angles are exactly 60°.

Isosceles

At least two sides are equal in length, and the angles opposite those sides are also equal.

Scalene

All three sides have different lengths, and all interior angles have different measures.

Right Angled

Contains one interior angle that is exactly 90°, following the Pythagorean theorem.

Acute

All three interior angles are less than 90°. Most equilateral triangles are acute.

Obtuse

Contains one interior angle that is greater than 90°. Only one such angle can exist.

Private Calculations: All data stays on your local browser.

Overview & Capabilities

Solve any triangle with our comprehensive Triangle Calculator! Enter any 3 known values (Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side) to calculate all missing side lengths, interior angles ($A+B+C=180^\circ$), area, perimeter, inradius ($r$), and circumradius ($R$) applying the Law of Sines and Law of Cosines.

Tutorial

How to Use

01
Select your known triangle parameter configuration (SSS, SAS, ASA, AAS, or SSA).
02
Enter the 3 known values into the corresponding side or angle boxes.
03
Click 'Solve Triangle' to evaluate all missing geometric dimensions.
04
Review the Law of Sines and Law of Cosines derivation formulas.
05
Inspect the dynamically drawn, scaled SVG triangle matching your angles and sides.
Capabilities

Key Features

5 Standard Triangle Solvers: Solves SSS (3 sides), SAS (2 sides + included angle), ASA, AAS, and SSA configurations.
Ambiguous Case (SSA) Handler: Identifies whether SSA inputs produce 0 triangles (impossible), 1 triangle, or 2 distinct valid triangles.
Law of Sines & Cosines Detailed: Displays explicit mathematical steps applying \(c^2 = a^2 + b^2 - 2ab\cos(C)\) and \(\frac{a}{\sin A} = \frac{b}{\sin B}\).
Complete Triangle Properties: Computes Area, Perimeter, Semi-perimeter, Incircle Radius (inradius), and Circumcircle Radius (circumradius).
Interactive Scaled SVG Render: Renders an accurately proportioned graphic of the solved triangle.
Applications

Common Use Cases

Education: Verifying geometry homework and understanding triangle properties.
Construction & Roof Pitch: Calculating lengths and angles for rafters and slopes.
Navigation & Land Surveying: Solving navigational triangles and boundary measurements.
Graphic Design: Determining precise coordinates for triangular elements in layouts.
DIY Projects: Measuring material needs for triangular shelves or garden beds.
Guidance

Tips & Best Practices

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Rule of Three: You must provide at least three measurements, and at least one must be a side length.
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Angle Limit: The sum of any two angles must be less than 180 degrees.
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Side Limit: The sum of any two sides must be greater than the third side (Triangle Inequality).
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Precision: We provide results up to 4 decimal places for professional accuracy.
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Right Triangles: If one angle is 90°, you can also use the Pythagorean theorem (a² + b² = c²).
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Answers

Frequently Asked Questions

Q What is the Law of Cosines and when is it used?

The Law of Cosines states c² = a² + b² - 2ab·cos(C). It is used to solve triangles when all 3 sides are known (SSS) or when 2 sides and their included angle are known (SAS).

Q What is the Law of Sines?

The Law of Sines states a / sin(A) = b / sin(B) = c / sin(C). It is used when an angle and its opposite side are known alongside one other side or angle (ASA, AAS, SSA).

Q What is the Ambiguous Case (SSA) in trigonometry?

When two sides and a non-included angle are given (SSA), the geometric configuration can result in no valid triangle (if side is too short), exactly one right/obtuse triangle, or two distinct valid triangles (one acute, one obtuse).