Find Missing Side of Triangle Calculator: Three Formulas, One Goal
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A find missing side of triangle calculator takes three known measurements — some combination of sides and angles — and returns the unknown side length. Three formulas cover every possible case: the Pythagorean theorem for right triangles, the Law of Cosines when you know two sides and the angle between them, and the Law of Sines when you have a side-angle pair plus one more angle. Pick the right formula, plug in the numbers, and the missing side drops out in one or two algebraic steps.
Find Missing Side of Triangle Calculator: The Three Core Formulas
Every triangle problem reduces to one question: what combination of information do I have? The answer determines which formula to reach for. Here they are, stripped to essentials:
| Situation | Formula | When It Fails |
|---|---|---|
| Right triangle, two sides known | c = √(a² + b²) | Only works with a 90° angle |
| Two sides + included angle (SAS) | c² = a² + b² − 2ab·cos(C) | Angle must be between the two sides |
| One side + two angles (AAS/ASA) | b = a·sin(B)/sin(A) | Ambiguous case (SSA) may give two answers |
The Pythagorean theorem is a special case of the Law of Cosines — set C = 90° and the cos(C) term vanishes, leaving you with c² = a² + b². That's not trivia; it matters because if your triangle is almost a right angle (say 88°), you'll get a noticeably wrong answer using Pythagoras instead of cosines. Our Pythagorean theorem calculator handles pure right-triangle problems, but for anything else, you need the full Law of Cosines.
Worked Example: Right Triangle with Legs 5 and 12
A ladder leans against a wall. The base sits 5 feet from the wall, and the ladder reaches 12 feet up. How long is the ladder?
Given: a = 5, b = 12, angle C = 90°
Step 1: c² = 5² + 12² = 25 + 144 = 169
Step 2: c = √169 = 13 feet
Check: 5, 12, 13 is a Pythagorean triple, so the answer is exact — no rounding needed.
Now flip it. You know the ladder is 13 feet and the base is 5 feet from the wall — how high does it reach? Rearrange: b = √(13² − 5²) = √(169 − 25) = √144 = 12 feet. Same numbers, opposite direction. The hypotenuse calculator focuses specifically on this kind of problem.
Worked Example: SAS with Two Sides and a 37° Angle
Two hikers start from the same camp. One walks 8 km northeast, the other walks 11 km at a bearing that creates a 37° angle between their paths. How far apart are they?
Given: a = 8, b = 11, C = 37°
Step 1: c² = 8² + 11² − 2(8)(11)·cos(37°)
Step 2: c² = 64 + 121 − 176 × 0.7986 = 185 − 140.55 = 44.45
Step 3: c = √44.45 ≈ 6.667 km
Notice how the answer is smaller than either input side. That's because 37° is acute — the sides point roughly the same direction, so the gap between the endpoints is relatively small. Bump the angle to 120° and c jumps to about 16.5 km. The included angle dominates the result more than most students expect. For more practice with this formula, try the Law of Cosines calculator.
Worked Example: AAS with One Side and Two Angles
A surveyor stands at point A, sights a tower at point B (angle 42°), then walks to point C and sights the same tower (angle 73°). The distance AC is 200 meters. What's the distance from C to the tower (side BC)?
Given: side AC (opposite angle B) = 200 m, angle A = 42°, angle B = 73°
Step 1: angle C = 180° − 42° − 73° = 65°
Step 2: Apply Law of Sines — BC/sin(A) = AC/sin(B)
Step 3: BC = 200 × sin(42°)/sin(73°) = 200 × 0.6691/0.9563
Step 4: BC ≈ 139.96 meters
The Law of Sines needs care with the ambiguous case (SSA): when you know two sides and an angle not between them, two different triangles might satisfy the constraints. Our Law of Sines calculator flags that situation automatically. But AAS and ASA configurations — like this surveying problem — always produce a single unique triangle.
How to Find the Missing Side of a Triangle: Decision Flowchart
Students waste time picking the wrong method, grinding through algebra, then starting over. This flowchart eliminates that step:
| Question | If Yes | If No |
|---|---|---|
| Is it a right triangle? | Pythagorean Theorem | Next question ↓ |
| Do you know two sides + the angle between them? | Law of Cosines | Next question ↓ |
| Do you know a side + its opposite angle + one more angle? | Law of Sines | Next question ↓ |
| Do you know a side + its opposite angle + another side? | Law of Sines (check for ambiguous case!) | Not enough info — need more data |
Print that table or memorize the first two rows — they handle 80% of textbook and real-world triangle problems. The triangle calculator can solve all configurations if you want to enter everything at once instead of choosing a method manually.
Four Gotchas That Produce Wrong Answers
Degrees vs. Radians
cos(60) in radians is 0.5, but cos(60°) in degrees is also 0.5 — lucky coincidence. Try cos(37°) = 0.7986 vs. cos(37 radians) = 0.7654. That tiny difference flips a 6.667 km answer to 6.85 km. Always check your calculator's angle mode.
Using Pythagorean Theorem on Non-Right Triangles
A triangle with sides 7 and 10 and a 50° included angle has a third side of about 7.66 (by Law of Cosines). Using Pythagoras gives √(49 + 100) = 12.21 — off by 59%. If the problem doesn't explicitly say "right triangle," don't assume it is.
Wrong Angle in Law of Cosines
The angle C in c² = a² + b² − 2ab·cos(C) must be the angle between sides a and b — the included angle. Using the angle opposite side a instead gives a completely different (and wrong) third side. Sketch the triangle and label everything before computing.
Ignoring the Ambiguous Case (SSA)
Given a = 10, b = 7, and angle A = 30°, the Law of Sines gives two valid triangles. One has angle B ≈ 20.5° and the other has B ≈ 159.5°. If you only report one answer on a test, you lose half the credit. The triangle angle calculator can help you verify both solutions.
Quick Reference: Common Triangle Side Ratios
Before reaching for a calculator, check if your triangle matches one of these classic ratios. They show up constantly in homework, standardized tests, and construction:
| Triangle Type | Side Ratios | Example |
|---|---|---|
| 3-4-5 right triangle | 3 : 4 : 5 | 6, 8, 10 (scaled ×2) |
| 5-12-13 right triangle | 5 : 12 : 13 | 10, 24, 26 (scaled ×2) |
| 45-45-90 | 1 : 1 : √2 | 7, 7, 9.899 |
| 30-60-90 | 1 : √3 : 2 | 5, 8.66, 10 |
| Equilateral | 1 : 1 : 1 | All sides equal; all angles 60° |
A carpenter framing a roof with a 3-4-5 check doesn't need a calculator. But scale the numbers oddly — say 7.5 and 10 with a 90° angle — and now you need √(56.25 + 100) = √156.25 = 12.5. Still a 3-4-5 ratio (×2.5), but harder to spot by eye. The right triangle calculator catches these patterns instantly.
Why Some Inputs Don't Form a Triangle
Not every set of numbers produces a valid triangle. The triangle inequality theorem states that the sum of any two sides must exceed the third side. Try sides 2, 3, and 10 — the two shorter sides (2 + 3 = 5) can't bridge the gap to reach 10. The "triangle" would collapse into a straight line.
Similarly, angles must sum to exactly 180°. Enter 90° + 95° and you've already exceeded 180° before the third angle — no triangle exists. Our calculator catches these impossible configurations and tells you why they fail, rather than returning a garbage number. For angle-specific problems, the triangle angle calculator complements this tool by solving for unknown angles rather than sides.
