How to Find Sides of a Triangle – Pythagoras, Trig, Sine & Cosine Laws
How to Find Missing Sides of a Triangle – Which Formula to Use
Finding a missing triangle side is easier when you identify the known-data pattern before choosing a formula.
On this page
Start by Classifying the Triangle
The correct side formula depends on what you know. Before calculating, ask two questions: Is the triangle right-angled? and which sides/angles are known?
| Known | Use |
|---|---|
| Two sides of a right triangle | Pythagorean theorem |
| One side + acute angle in a right triangle | sin, cos or tan |
| SAS | Law of Cosines |
| ASA/AAS or known opposite pair | Law of Sines |
| SSS | No side is missing; use Law of Cosines to find angles |
Pythagorean Theorem — Right Triangles
c is the hypotenuse, always opposite the right angle.
Example: legs 9 and 12:
If c and one leg are known, rearrange: a=√(c²−b²).
Trigonometry — Right Triangle with an Angle
Example: hypotenuse 20 m and angle 35°; find opposite side:
Law of Sines
Use this when you know at least one complete opposite side-angle pair.
Example: A=40°, B=70°, a=8:
Keep each side paired with its opposite angle.
Law of Cosines
Use this for SAS when two sides and their included angle are known.
Example: a=6, b=9, C=50°:
Worked Examples
Find hypotenuse: legs 7 and 24
Find a leg: c=17, b=8
AAS: A=50°, C=60°, a=10
B=70°. Then c=10·sin60°/sin50°≈11.305.
SAS: sides 10 and 13 with included angle 42°
Common Mistakes
Practice Questions
Right triangle: c=10, one leg=6
Answer: missing leg = √(100−36)=8.
SAS: a=5, b=8, C=90°
Answer: Law of Cosines reduces to Pythagoras: c=√89≈9.434.
Questions People Ask
Open any question for a detailed explanation, not just a one-line answer.
How do I find a missing side in a right triangle?
Use Pythagoras if two sides are known. If one side and an acute angle are known, use sine, cosine or tangent depending on the side relationship.
Which side is the hypotenuse?
The hypotenuse is the side opposite the 90° angle. It is always the longest side in a right triangle.
How do I find a side if the triangle is not right-angled?
Use the Law of Sines when an opposite side-angle pair is known, or the Law of Cosines for SAS/SSS situations.
When should I use the Law of Cosines?
Use it when two sides and the included angle are known (SAS), or when all three sides are known and you need an angle.
When should I use the Law of Sines?
Use it when you know a side and its opposite angle, plus another angle or side you want to relate.
Why can’t I always use Pythagoras?
Pythagoras depends on a 90° angle. Without that right angle, the missing cosine term in the more general Law of Cosines is no longer zero.
How do I check the missing side?
Verify the triangle inequality and confirm the longest side lies opposite the largest angle. For right triangles, substitute back into a²+b²=c².
Can the Triangle Calculator choose the method for me?
Yes. The Triangle Calculator lets you choose what information is known and provides method-specific working.
Related CalcTypes Resources
Methodology
This guide maps side-solving formulas to known-data patterns and includes validation rules so formula selection is based on geometry rather than letter matching.