Solving the Triangle You Can't Quite Figure Out
Whether you're working with a right triangle (where one angle is 90 degrees) or an oblique triangle (all angles different), triangles have hidden relationships. Know a few measurements-sides, angles, or a mix-and a triangle calculator can find everything else. You don't need to remember the law of sines or law of cosines; the calculator applies the right tool automatically.
What This Calculator Does
A triangle calculator finds missing sides and angles using the relationships that connect all triangle measurements. For right triangles, it uses the Pythagorean theorem. For any triangle, it applies the law of sines and law of cosines. Give it at least three pieces of information (usually two sides and an angle, or two angles and a side), and it solves for everything remaining. You also get the area using Heron's formula or standard methods.
How to Use This Calculator
Input the measurements you know. Most triangle calculators ask for three values (since a triangle is defined by three independent measurements). You might enter two sides and the angle between them, or two angles and a side, or all three sides. The calculator works backward and forward through triangle geometry to find what's missing.
Some calculators visually show the triangle you're describing, so you can verify you've entered the right values. The output displays all three sides, all three angles, the area, and sometimes the perimeter and inradius (radius of the inscribed circle).
The calculator works in degrees or radians, depending on your preference. For real-world problems, degrees are more intuitive; for trigonometry and calculus, radians are standard.
The Formula Behind the Math
Pythagorean Theorem (right triangles only): c² = a² + b²
Where c is the hypotenuse (longest side, opposite the right angle) and a, b are the other two sides.
Example: a = 3, b = 4
c² = 9 + 16 = 25
c = 5
Law of Sines (any triangle): a/sin(A) = b/sin(B) = c/sin(C)
Where a, b, c are sides and A, B, C are the opposite angles. This is useful when you know one side and its opposite angle, and want to find another side.
Example: If side a = 10, angle A = 30°, and angle B = 45°, find side b.
10/sin(30°) = b/sin(45°)
10/0.5 = b/0.707
20 = b/0.707
b ≈ 14.14
Law of Cosines (any triangle): c² = a² + b² - 2ab×cos(C)
This generalizes the Pythagorean theorem to non-right triangles. When C = 90°, cos(90°) = 0, and it reduces to the Pythagorean theorem.
Example: Two sides a = 5, b = 7, angle C = 60° between them.
c² = 25 + 49 - 2(5)(7)cos(60°)
c² = 74 - 70(0.5)
c² = 74 - 35 = 39
c ≈ 6.24
Area (Heron's formula): s = (a+b+c)/2; Area = √(s(s-a)(s-b)(s-c))
First calculate the semi-perimeter s, then the area. Works for any triangle given all three sides.
Example: sides 3, 4, 5 (right triangle)
s = (3+4+5)/2 = 6
Area = √(6 × 3 × 2 × 1) = √36 = 6 square units
Alternative for triangles with two sides and included angle:
Area = ½ × a × b × sin(C)
Our calculator does all of this instantly-but now you understand exactly what it's computing.
Land Surveying and Property Boundaries
A surveyor measures a triangular property: one side is 100 feet, the adjacent side is 85 feet, and the angle between them is 75 degrees. What's the third side (opposite this angle)?
Using the law of cosines:
c² = 100² + 85² - 2(100)(85)cos(75°)
c² = 10,000 + 7,225 - 17,000(0.2588)
c² = 17,225 - 4,399.6 ≈ 12,825.4
c ≈ 113.25 feet
The property's third boundary is about 113 feet. A triangle calculator saves surveyors from hand calculations and potential errors.
Roof Pitch and Construction
A house needs a roof with a specific slope. If the horizontal span is 30 feet and the roof pitch creates a triangle where the height is 10 feet, what's the slant length (the actual material needed)?
Using the Pythagorean theorem (the roof is a right triangle):
c² = 30² + 10²
c² = 900 + 100 = 1,000
c ≈ 31.62 feet
The roofer needs 31.62 feet of materials for each slant side (twice that for both slopes). A triangle calculator removes errors in this critical calculation.
Navigation and Triangulation
A sailor knows the distance to two lighthouses and the angle between them. Light A is 5 miles away, Light B is 7 miles away, and the angle between them (at the ship) is 65 degrees. How far apart are the lighthouses?
Using the law of cosines:
c² = 5² + 7² - 2(5)(7)cos(65°)
c² = 25 + 49 - 70(0.4226)
c² = 74 - 29.58 ≈ 44.42
c ≈ 6.67 miles
The lighthouses are approximately 6.67 miles apart. Navigation and survey problems frequently use triangle relationships.
Tips and Things to Watch Out For
Right triangle methods don't work on oblique triangles. The Pythagorean theorem applies only to the one triangle in a million with a 90-degree angle. For general triangles, use the law of sines or law of cosines.
Angle sum is always 180 degrees. If you calculate one angle and know another, the third is 180° minus the sum of the two known angles. Use this as a sanity check on your calculator's results.
Law of sines has an ambiguous case. When given two sides and an angle opposite one of them (SSA), there might be zero, one, or two possible triangles. A good calculator warns you about this. Manual calculation requires careful checking.
Angle units matter. Calculators use either degrees or radians. Most real-world problems use degrees. Mixing units (entering one angle in degrees and another in radians) produces nonsense. Confirm your calculator's angle unit setting.
Round intermediate steps carefully. If you calculate a side then use it to find an angle, small rounding errors compound. A calculator using full precision throughout avoids this problem.
Verify the triangle is possible. The sum of any two sides must exceed the third side (triangle inequality). If your calculator accepts sides 2, 3, and 10, something is wrong—2 + 3 < 10, so no triangle exists.
Frequently Asked Questions
What's the difference between a right triangle and an oblique triangle?
A right triangle has one 90-degree angle. An oblique triangle has no 90-degree angles. Right triangles use simpler formulas (Pythagorean theorem). Oblique triangles require the law of sines or law of cosines.
Can I use the Pythagorean theorem on non-right triangles?
No. The Pythagorean theorem applies only to right triangles. For other triangles, it's replaced by the law of cosines, which reduces to the Pythagorean theorem when the angle is 90 degrees.
How do I know which law to use-sines or cosines?
Use the law of sines when you know a side and its opposite angle. Use the law of cosines when you know two sides and the included angle (angle between them), or all three sides. Most calculators choose automatically.
What's the ambiguous case in the law of sines?
When given two sides and an angle opposite one (SSA), sometimes two different triangles fit the measurements. A calculator should warn you if this occurs. Manual calculation requires checking both possibilities.
What's the semi-perimeter in Heron's formula?
It's half the perimeter (half the sum of all three sides). Heron's formula uses it as an intermediate calculation step to find the area when you know all three sides but not the height.
Can I find the area without knowing a height?
Yes. If you know all three sides, use Heron's formula. If you know two sides and the included angle, use Area = ½ × a × b × sin(C). A triangle calculator handles both approaches.
What's an inradius and why does it matter?
The inradius is the radius of the circle that fits inside the triangle touching all three sides. It's useful in advanced geometry, construction, and design. Most basic triangle calculators don't compute it, but advanced ones do.
Related Calculators
The Pythagorean theorem calculator focuses specifically on right triangle calculations. The circle calculator helps with problems involving triangles inscribed in or circumscribing circles. The area calculator verifies triangle areas independently if needed.