Triangle Calculator

Related Calculators

Known values: The input pattern you choose so the calculator knows which triangle rule to use.

Side a, side b, side c: The three side lengths. Each side is opposite the angle with the same letter.

Angle A, Angle B, Angle C: The three interior angles of the triangle, measured in degrees.

Number of valid triangles: How many triangles fit your inputs. Most cases give 1, but SSA can give 0, 1, or 2.

Perimeter: The total distance around the triangle.

Semiperimeter: Half of the perimeter. It is often used to find area.

Area: The amount of space inside the triangle, measured in square units.

Altitude to side a: A perpendicular height to side a from the opposite vertex. The same idea applies to sides b and c.

Median to side a: A segment from angle A to the midpoint of side a. The same idea applies to sides b and c.

Inradius: The radius of the circle that fits inside the triangle and touches all three sides.

Circumradius: The radius of the circle that passes through all three vertices of the triangle.

Mistake: Picking the wrong Known values pattern, such as using SSA when your angle is actually between the two known sides.
Fix: Match the pattern first. Use 2 sides and included angle (SAS) only when the known angle is between the two known sides.

Mistake: Entering a side under the wrong label, like putting the length opposite angle C into Side a (units) .
Fix: Keep the naming consistent: Solved side a is always opposite Solved angle A , and the same rule applies for b and c.

Mistake: Typing invalid angle values in Angle A (degrees) , Angle B (degrees) , or Angle C (degrees) , such as 0, 180, or angles that add to 180 before the missing angle is solved.
Fix: Each entered angle must be greater than 0 and less than 180, and in the two-angle mode the known angles must add to less than 180.

Mistake: Using impossible side lengths for SSS, such as values where one side is at least as long as the other two combined.
Fix: For 3 sides (SSS) , make sure each side is less than the sum of the other two before you calculate Number of valid triangles .

Mistake: Mixing units after choosing Length unit , like entering one side in feet and another in inches.
Fix: Convert all lengths first, then enter every side in one consistent unit so Perimeter , Area , and the altitude outputs stay meaningful.

Mistake: Thinking an SSA setup should always give one answer.
Fix: Check Number of valid triangles . In SSA, the result can be 0, 1, or 2 valid triangles depending on the side-angle relationship.

• This calculator works for plane triangles in standard Euclidean geometry, not spherical or 3D triangles.
• All side lengths must be positive, and any entered angle must be greater than 0 degrees and less than 180 degrees.
• For the two-angle mode, the entered angles must add to less than 180 degrees so a third angle can exist.
• For SSS, the side lengths must satisfy the triangle inequality: each side must be less than the sum of the other two sides.
• SSA can produce 0, 1, or 2 valid triangles. The calculator shows all valid solutions, but small rounding differences near the boundary can make a case switch between 1 and 2 solutions.
• Area, altitudes, medians, inradius, and circumradius are derived from the solved triangle, so they depend on the accuracy of your original inputs.
• Results are rounded to the selected Decimal places, so angle sums and related checks may be off by a tiny amount after rounding.

How the calculator solves the triangle

The calculator first checks which Known values mode you selected, then uses the matching geometry rule. It ignores hidden fields so only the visible inputs affect the answer.

Main formulas

A + B + C = 180 degrees

a^2 = b^2 + c^2 - 2bc cos(A)

a / sin(A) = b / sin(B) = c / sin(C)

P = a + b + c

s = P / 2

Area = sqrt(s(s-a)(s-b)(s-c))

h_a = 2Area / a

m_a = 0.5 x sqrt(2b^2 + 2c^2 - a^2)

r = Area / s

R = a / (2 sin(A))

By input pattern

SSS: If you know all three sides, the calculator checks the triangle inequality, then uses the Law of Cosines to find all three angles.

SAS: If you know two sides and the included angle, the calculator first finds the missing side with the Law of Cosines, then finds the other angles.

ASA or AAS: If you know two angles and one side, the calculator finds the missing angle from the 180 degree angle sum, then uses the Law of Sines to find the missing sides.

SSA: If you know two sides and a non-included angle, the calculator uses the Law of Sines and checks the ambiguous case. That means the inputs may create no triangle, one triangle, or two triangles [2].

Derived measurements

After the sides and angles are solved, the calculator finds Perimeter and Semiperimeter. It then uses Heron's formula to compute Area from the three sides [1]. Altitudes come from area divided by the matching base, medians come from the standard median-length formula, the Inradius comes from area divided by semiperimeter, and the Circumradius comes from a side-angle pair.

Worked mini-example

Suppose you use SSS with side lengths 3, 4, and 5. The perimeter is 12, so the semiperimeter is 6.

Area = sqrt(6 x (6-3) x (6-4) x (6-5)) = sqrt(36) = 6

The altitude to side c is 2 x 6 / 5 = 2.4. The inradius is 6 / 6 = 1. This matches the familiar 3-4-5 right triangle.

Error checks and edge handling

The calculator blocks non-positive side lengths, invalid angles, impossible SSS triangles, and angle combinations that cannot make a triangle. In SSA mode, it clamps tiny floating-point drift before inverse trig steps, but if the geometry is actually impossible, it returns 0 valid triangles instead of forcing a false answer.

Assumptions behind the results

The math assumes a flat plane triangle, consistent length units, and correctly matched opposite side-angle labels. If your measurements are rounded or entered under the wrong labels, the solved triangle and all derived outputs can change.