The method you use to find a missing side of a triangle depends on what information you already have. If you know two sides of a right triangle, the Pythagorean theorem gives you the third. If you’re working with a non-right triangle, the Law of Sines or the Law of Cosines will get you there, depending on whether you have angle-side pairs or a side-angle-side setup. Here’s how each method works, step by step.
Right Triangles: The Pythagorean Theorem
If your triangle has a 90-degree angle, the Pythagorean theorem is the fastest path to the missing side. The formula relates the two shorter sides (called legs) to the longest side (called the hypotenuse, which is always opposite the right angle):
a² + b² = c²
Here, a and b are the legs and c is the hypotenuse.
Finding the Hypotenuse
If you know both legs, plug them into the formula and solve for c. For example, if one leg is 5 and the other is 12:
5² + 12² = c²
25 + 144 = c²
169 = c²
c = √169 = 13
Finding a Missing Leg
If you know the hypotenuse and one leg, rearrange the formula to isolate the unknown leg. Say the hypotenuse is 10 and one leg is 6:
6² + b² = 10²
36 + b² = 100
b² = 100 − 36 = 64
b = √64 = 8
Since you’re solving for a length, you only need the positive square root.
Special Right Triangles: Built-In Shortcuts
Two types of right triangles have fixed side ratios, which means you can find a missing side without running the full Pythagorean theorem if you recognize the angle pattern.
A 45-45-90 triangle has two equal legs and a hypotenuse that is √2 times the length of either leg. The ratio is x : x : x√2. So if one leg is 7, the other leg is also 7 and the hypotenuse is 7√2 (about 9.9).
A 30-60-90 triangle has sides in the ratio x : x√3 : 2x. The shortest side (opposite the 30-degree angle) is x, the longer leg (opposite the 60-degree angle) is x√3, and the hypotenuse is 2x. If the shortest side is 4, the longer leg is 4√3 (about 6.93) and the hypotenuse is 8.
These shortcuts save time on standardized tests and homework problems where the angles are explicitly 30, 45, 60, or 90 degrees.
Non-Right Triangles: Law of Sines
When a triangle has no right angle, you need a different approach. The Law of Sines works when you know at least one side and its opposite angle, plus one additional angle or side.
The formula is:
a / sin(A) = b / sin(B) = c / sin(C)
Each lowercase letter is a side, and each uppercase letter is the angle opposite that side.
The most straightforward case is when you know two angles and one side. Since the three angles of any triangle add up to 180 degrees, knowing two angles automatically gives you the third. From there, set up a proportion. For example, suppose angle A is 40°, angle B is 75°, and side a (opposite angle A) is 10:
10 / sin(40°) = b / sin(75°)
b = 10 × sin(75°) / sin(40°)
b = 10 × 0.9659 / 0.6428
b ≈ 15.03
The Law of Sines also works when you know two sides and an angle opposite one of them, though this setup can sometimes produce two valid triangles (a situation called the “ambiguous case”). If your calculator gives you an answer and the numbers seem off, check whether a second solution exists by seeing if the supplementary angle also fits within the 180-degree total.
Non-Right Triangles: Law of Cosines
The Law of Cosines is your tool when the known angle sits between the two known sides (a side-angle-side configuration). The Law of Sines can’t handle this setup because the known angle doesn’t correspond to a known side.
The formula is:
c² = a² + b² − 2ab × cos(C)
Here, C is the angle between sides a and b, and c is the side you’re solving for. For example, if side a is 8, side b is 11, and the included angle C is 37°:
c² = 8² + 11² − 2(8)(11) × cos(37°)
c² = 64 + 121 − 176 × 0.7986
c² = 185 − 140.55
c² = 44.45
c ≈ 6.67
You can also use the Law of Cosines when you know all three sides and need to find an angle, then use that angle to solve further. Rearranging for the angle looks like this:
cos(C) = (a² + b² − c²) / 2ab
Using Area to Find a Missing Side
Sometimes a problem gives you the area of the triangle instead of a third side or angle. In that case, work backward from an area formula.
The basic area formula is:
Area = (base × height) / 2
If you know the area and the height, solve for the base: base = 2 × Area / height. If you know the area and the base, solve for the height the same way.
For non-right triangles where you know two sides and the included angle, the area formula becomes:
Area = (a × b × sin(C)) / 2
If the area is given along with one side and the included angle, you can rearrange to find the other side. Say the area is 30, side a is 10, and angle C is 50°:
30 = (10 × b × sin(50°)) / 2
60 = 10 × b × 0.7660
60 = 7.66b
b ≈ 7.83
How to Pick the Right Method
The information you’re given determines which formula to use. Here’s a quick guide:
- Right triangle, two sides known: Pythagorean theorem (a² + b² = c²)
- Right triangle with 30-60-90 or 45-45-90 angles: Use the fixed side ratios for a faster solve
- Two angles and one side (AAS or ASA): Law of Sines
- Two sides and the angle between them (SAS): Law of Cosines
- Two sides and an angle opposite one of them (SSA): Law of Sines, but check for the ambiguous case
- Three sides, no angles (SSS): Law of Cosines to find an angle first, then Law of Sines for the rest
- Area plus partial side/angle info: Rearrange the appropriate area formula
Start by labeling what you know. Write down every given side length, angle, and any other information like area. Then match your known quantities to the list above. The right formula will be obvious once you see what’s missing.