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Triangle Calculator

Solve any triangle — find all sides, angles, area, and perimeter using SSS, SAS, or AAS

Triangle Calculator

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Quick summary

A triangle calculator uses trigonometric laws to find all unknown sides and angles. Given any 3 independent pieces of information (SSS, SAS, AAS, ASA), the calculator solves for everything else using the Law of Cosines and Law of Sines.

Reviewed by Simple Calculators TeamLast reviewed

How to use this calculator

  1. Enter the inputs requested by the calculator.
  2. Choose the units or options that match your situation and review the assumptions.
  3. Run or update the calculation to see the result.
  4. Review the result, breakdown, and formulas shown on the page.
  5. Check the inputs, units, dates, and limitations before saving or sharing the result.

Triangle Formulas & Laws

Law of Cosines (SSS/SAS)

c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab \cos(C)
a, b, c
The three sides of the triangle
C
The angle opposite side c

Worked example

Triangle with a=3, b=4, c=5

Given
a = 3
b = 4
c = 5
Calculation
C=arccos⁡(9+16−2524)=arccos⁡(0)C = \arccos\left(\frac{9+16-25}{24}\right) = \arccos(0)
Result
C = 90° (right triangle)

Law of Sines (AAS/ASA)

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
a, b, c
Sides opposite angles A, B, C

Worked example

A=45°, B=60°, a=10

Given
A = 45°
B = 60°
a = 10
Calculation
b=10×sin⁡60°sin⁡45°b = 10 \times \frac{\sin 60°}{\sin 45°}
Result
≈ 12.247

Triangle Area

A=12absin⁡(C)A = \frac{1}{2} a b \sin(C)
a, b
Two sides of the triangle
C
The angle between those two sides

Worked example

a=5, b=7, C=60°

Given
a = 5
b = 7
C = 60°
Calculation
A=12×5×7×sin⁡(60°)A = \frac{1}{2} \times 5 \times 7 \times \sin(60°)
Result
≈ 15.155

Frequently asked questions

What is the Law of Cosines?

The Law of Cosines generalizes the Pythagorean theorem: c² = a² + b² − 2ab×cos(C). It works for any triangle (not just right triangles) and is used when you know SSS or SAS.

When do I use the Law of Sines?

Use the Law of Sines when you know two angles and a side (AAS or ASA): a/sin(A) = b/sin(B) = c/sin(C). It cannot be used for SSS or SAS directly.

How do I identify the type of triangle?

Equilateral: all 3 sides equal. Isosceles: exactly 2 sides equal. Scalene: no sides equal. Right: one 90° angle (a²+b²=c²). Obtuse: one angle > 90°. Acute: all angles < 90°.

Why does my triangle fail the triangle inequality?

For a valid triangle, each side must be less than the sum of the other two (a < b+c). If not, the three lengths cannot form a closed triangle.

Key terms

SSS
Side-Side-Side: all three sides are known.
SAS
Side-Angle-Side: two sides and the included angle are known.
AAS
Angle-Angle-Side: two angles and a non-included side are known.
Triangle Inequality
The sum of any two sides must be greater than the third side.

Methodology, sources & limitations

This page shows the formula, assumptions, and limitations used for this estimate. Rates, thresholds, and other time-sensitive inputs can change, so verify them against the relevant official source before relying on a result.

For our review approach and the primary sources we consult, see the methodology and source policy. If you spot an outdated figure or formula, please contact us with a correction.