Triangle Input Parameters

Free triangle calculator - Enter any 3 values (with at least one side) to solve any triangle. Works with right triangles, isosceles triangles, scalene triangles, and equilateral triangles using SSS, SAS, ASA, AAS, or SSA methods.

How to Solve Triangles: SSS, SAS, ASA, AAS, SSA Explained

What is SSS, SAS, ASA, AAS, SSA?

These abbreviations describe which parts of a triangle you know:

  • S = Side (length of a triangle edge)
  • A = Angle (measure in degrees)

The order matters! It tells you if elements are adjacent (next to each other) or not.

📐 SSS - Three Sides Known

Example: Side A = 5, Side B = 7, Side C = 9

How to solve:

  1. Use Law of Cosines to find first angle
  2. Use Law of Cosines or Law of Sines for second angle
  3. Third angle = 180° - angle1 - angle2

Always has exactly one solution if sides form valid triangle.

📐 SAS - Two Sides and Included Angle

Example: Side A = 5, Angle C = 60°, Side B = 7

How to solve:

  1. Use Law of Cosines to find the third side
  2. Use Law of Sines to find second angle
  3. Third angle = 180° - given angle - angle2

The angle must be between the two known sides!

📐 ASA - Two Angles and Included Side

Example: Angle A = 45°, Side C = 10, Angle B = 60°

How to solve:

  1. Find third angle: C = 180° - 45° - 60° = 75°
  2. Use Law of Sines to find other two sides

The side must be between the two known angles!

📐 AAS - Two Angles and Non-Included Side

Example: Angle A = 45°, Angle B = 60°, Side A = 8

How to solve:

  1. Find third angle: C = 180° - 45° - 60° = 75°
  2. Use Law of Sines to find remaining sides

Similar to ASA, always has one solution.

⚠️ SSA - The Ambiguous Case

Example: Side A = 7, Side B = 5, Angle A = 40°

Why ambiguous? Can have 0, 1, or 2 valid triangles!

  1. Use Law of Sines to find second angle
  2. Check if solution exists (sin value ≤ 1)
  3. May have two possible angles (obtuse or acute)
  4. Verify triangle inequality for each case

⚠️ Always check both possible solutions!

❌ AAA - Cannot Solve

Example: Angle A = 60°, Angle B = 70°, Angle C = 50°

Why can't we solve?

These angles describe the shape but not the size. Infinite triangles exist with these angles - from tiny to huge!

✅ Solution: You need at least ONE side length to determine the exact triangle.

💡 Quick Tips

  • Always check: Sum of angles = 180°
  • Triangle inequality: Any side < sum of other two sides
  • Need at least one side to find exact triangle size
  • SSA is tricky: Always verify your answer makes sense
  • Right triangles: Can use Pythagorean theorem when angle = 90°

Triangle Mathematics Reference

Triangle Solver Methods Explained

  • SSS (Side-Side-Side): When you know all three side lengths. Use Law of Cosines to find angles. Works for all triangle types.
  • SAS (Side-Angle-Side): When you know two sides and the angle between them. Use Law of Cosines to find the third side, then Law of Sines for remaining angles.
  • ASA (Angle-Side-Angle): When you know two angles and the side between them. Calculate third angle (180° - A - B), then use Law of Sines to find other sides.
  • AAS (Angle-Angle-Side): When you know two angles and a non-included side. Calculate third angle first, then use Law of Sines to find remaining sides.
  • SSA (Side-Side-Angle): When you know two sides and a non-included angle. May have 0, 1, or 2 solutions (ambiguous case). Use Law of Sines carefully.
  • AAA (Angle-Angle-Angle): Cannot solve - infinite similar triangles possible. Need at least one side length.

Key Formulas

Law of Cosines (for SSS and SAS triangles):
$$c^2 = a^2 + b^2 - 2ab \cos(C)$$
$$a^2 = b^2 + c^2 - 2bc \cos(A)$$
$$b^2 = a^2 + c^2 - 2ac \cos(B)$$
Law of Sines (for ASA, AAS, and SSA triangles):
$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R$$

where R is the circumradius

Angle Sum Property:
$$A + B + C = 180° = \pi \text{ radians}$$
Area Formulas:
$$\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B)$$
$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

where $s = \frac{a+b+c}{2}$ (Heron's formula)

Triangle Types Our Calculator Solves

By Angles:
  • Acute Triangle: All angles $< 90°$
  • Right Triangle: One angle $= 90°$ (Pythagorean theorem)
  • Obtuse Triangle: One angle $> 90°$
By Sides:
  • Equilateral Triangle: $a = b = c$, all angles $= 60°$
  • Isosceles Triangle: Two sides equal - perfect for isosceles triangle calculations
  • Scalene Triangle: All sides different lengths

When to Use This Triangle Calculator

  • Solve homework and math problems with step-by-step formulas
  • Engineering surveying and structural design
  • Right triangle calculations using Pythagorean theorem
  • Find missing sides and angles in any triangle
  • Calculate triangle area and perimeter
  • Navigation, GPS, and trigonometry applications
  • Architecture and construction planning

Advanced Triangle Calculator Features

Special Right Triangles

45-45-90 Triangle:
$$\text{Sides ratio: } 1 : 1 : \sqrt{2}$$
$$\text{If legs = } a, \text{ then hypotenuse = } a\sqrt{2}$$
30-60-90 Triangle:
$$\text{Sides ratio: } 1 : \sqrt{3} : 2$$
$$\text{If short leg = } a, \text{ then long leg = } a\sqrt{3}$$
$$\text{hypotenuse = } 2a$$

Circle Relationships

Circumradius (R):
$$R = \frac{a}{2\sin(A)} = \frac{b}{2\sin(B)} = \frac{c}{2\sin(C)}$$
$$R = \frac{abc}{4 \cdot \text{Area}}$$
Inradius (r):
$$r = \frac{\text{Area}}{s} = \frac{\text{Area}}{\frac{a+b+c}{2}}$$
$$r = (s-a)\tan\left(\frac{A}{2}\right) = (s-b)\tan\left(\frac{B}{2}\right) = (s-c)\tan\left(\frac{C}{2}\right)$$

Triangle Inequalities

Basic Triangle Inequality:
$$a + b > c \quad \text{and} \quad a + c > b \quad \text{and} \quad b + c > a$$
Angle Relationships:
$$\text{Largest angle is opposite longest side}$$
$$\text{If } a > b > c, \text{ then } A > B > C$$

Advanced Area Formulas

$$\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B)$$
$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \text{ (Heron's)}$$
$$\text{Area} = rs \text{ where } r \text{ is inradius, } s \text{ is semiperimeter}$$
$$\text{Area} = \frac{abc}{4R} \text{ where } R \text{ is circumradius}$$