Triangle Calculator
Solve any triangle from any combination of sides and angles using SSS, SAS, ASA, AAS, or right triangle methods. Get all sides, angles, area, perimeter, and more.
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How It Works
With all three sides known, use the Law of Cosines to find each angle. Then compute area using Heron's formula with semiperimeter s = (a+b+c)/2.
cos(A)=(b²+c²-a²)/(2bc), Area=√(s(s-a)(s-b)(s-c))a=3,b=4,c=5 → A=37°,B=53°,C=90°Find the third side using Law of Cosines: c²=a²+b²-2ab·cos(C). Then use Law of Sines to find remaining angles.
c=√(a²+b²-2ab·cos(C))a=5,C=60°,b=7 → c=6.24Find the third angle: C=180°-A-B. Use the Law of Sines to find remaining sides: a=c·sin(A)/sin(C).
a=c·sin(A)/sin(C)A=45°,c=8,B=60° → C=75°Third angle: C=180°-A-B. Use Law of Sines with the known side to find the others.
b=a·sin(B)/sin(A)A=30°,B=60°,a=5 → b=8.66,c=10With two values known (any combination of legs a, b or hypotenuse c): a²+b²=c². Angles from inverse trig: A=arctan(a/b).
a²+b²=c², A=arctan(a/b)a=3,b=4 → c=5, A=36.87°Inradius r = Area/s (s=semiperimeter) — radius of inscribed circle. Circumradius R = abc/(4·Area) — radius of circumscribed circle.
r=Area/s, R=abc/(4·Area)3-4-5 triangle: r=1, R=2.5Quick Reference
Common examples — verify instantly above.
SSS
3-4-5
Right triangle, A=37°
SSS
5-5-5
Equilateral, 60° each
Right
a=3, b=4
c=5, A=36.87°
Right
a=5, c=13
b=12, A=22.6°
Area
base=6, h=4
Area=12
Perimeter
3-4-5
Perimeter=12
Inradius
3-4-5
r=1
Circumradius
3-4-5
R=2.5
Tips & Shortcuts
For a right triangle, the hypotenuse is always opposite the 90° angle and is the longest side.
Law of Sines (a/sinA = b/sinB) works best when two angles and a side are known.
Law of Cosines works for any triangle. It is the Pythagorean theorem generalized to non-right triangles.
The sum of all interior angles of any triangle always equals 180°.
An equilateral triangle has all sides equal and all angles equal to 60°.
Heron's formula for area requires only three sides — no height needed.
Common Mistakes
Thinking the sum of angles must be 180 radians instead of degrees
Interior angles sum to 180° (degrees), not 180 radians. Always work in degrees unless specifically using radians.
Using a side when the formula needs its opposite angle
Law of Sines: side a is opposite angle A. Always match each side with its opposite angle.
Forgetting that AAS can give two triangles (ambiguous case)
AAS gives a unique triangle. But SSA (not covered here) can give 0, 1, or 2 triangles depending on the values.
Computing perimeter as area or vice versa
Perimeter = a+b+c (linear units). Area uses square units. They have different formulas and different units.
Using slant sides instead of perpendicular height for area
Area = base × perpendicular height / 2. The height must be at 90° to the base, not a slanting side.
Rounding intermediate angles before finding sides
Keep full precision in intermediate steps. Round only the final answer to avoid accumulated errors.
Frequently Asked Questions
SSS means three sides are known. Use the Law of Cosines to find all angles: cos(A) = (b²+c²-a²)/(2bc).
SAS means two sides and the included angle between them are known. Use the Law of Cosines to find the third side, then Law of Sines for remaining angles.
ASA means two angles and the side between them are known. The third angle = 180°-A-B. Use the Law of Sines to find remaining sides.
a/sin(A) = b/sin(B) = c/sin(C). The ratio of each side to the sine of its opposite angle is constant for any triangle.
c² = a² + b² - 2ab×cos(C). It generalizes the Pythagorean theorem to non-right triangles.
The inradius is the radius of the largest circle that fits inside the triangle. inradius = Area / semiperimeter.
Law of Sines (a/sin A = b/sin B = c/sin C): use for ASA (two angles + included side), AAS (two angles + non-included side), or SSA (two sides + non-included angle — the ambiguous case may give 0, 1, or 2 solutions). Law of Cosines (c²=a²+b²−2ab·cos C): use for SSS (all three sides) or SAS (two sides + included angle). Law of Cosines reduces to the Pythagorean theorem when C = 90°.
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