Triangle Calculator

How to use the triangle calculator

Choose a mode before entering measurements

SSS uses three sides. SAS uses two sides and the angle between them. ASA uses two angles and their included side, while AAS uses two angles and a non-included side. SSA handles two sides with an angle that is not between them and must be treated separately.

1. 1

   SSS

   Check the triangle inequality, then derive stable area and all three angles.

2. 2

   SAS

   Use the included angle to calculate the opposite side and area.

3. 3

   ASA or AAS

   Find the third angle by subtraction and scale the sides with the law of sines.

4. 4

   SSA

   Solve a quadratic for every positive third-side candidate and validate each triangle.

5. 5

   Base and height

   Return the area while leaving unsupported sides and angles undetermined.

Three side lengths must pass the triangle inequality: the sum of any two sides must exceed the third. Lengths 2, 3 and 6 fail because 2 + 3 ≤ 6. Rather than allowing a square root to become NaN, the calculator returns zero area and explains that the lengths cannot close.

For sides 7, 7 and 0.00000002, the exact area used here is 0.00000007. The textbook semiperimeter calculation returns 0.00000006999999957 in ordinary floating-point arithmetic, while the stable rearrangement returns 0.00000007 without losing the seventh significant figure.

The SSA solver treats the third side as the unknown in a law-of-cosines quadratic. This avoids relying on inverse sine near its worst-conditioned boundary. For A = 30°, a = 6 and b = 12, the discriminant is zero and the single result has B = 90°, not a slightly acute approximation.

For a completed triangle, two angles come from the law of cosines and the third is calculated as 180° minus their sum. This prevents three independently rounded inverse-cosine results from displaying an angle total such as 59.99° + 60.00° + 60.00°.

Base and perpendicular height determine area but not the triangle. The apex may slide along a line parallel to the base while preserving both measurements. Side lengths, angles, perimeter, inradius and circumradius therefore remain unavailable rather than being filled with assumed values.

Completed results also include height to side a, inradius, circumradius and two classifications. The angle classification is right, acute or obtuse. The side classification is equilateral, isosceles or scalene. These labels describe the solved geometry and are not inferred for base-and-height-only input.

Why SSA can produce two triangles

With A and its opposite side a fixed, another side b can sometimes swing into two positions. The corresponding angle B is acute in one position and its obtuse supplement in the other. Both candidates must still leave a positive third angle.

For A = 30°, a = 8 and b = 12, the resulting B values are 48.59° and 131.41°. The first triangle is obtuse at C = 101.41°, while the second is obtuse at B = 131.41°. Neither completed triangle is acute.

Why the SSS area formula is rearranged

The familiar semiperimeter form is mathematically correct, but its floating-point evaluation can lose precision when one side is tiny compared with the other two. Intermediate subtraction then removes matching leading digits before the remaining factors are multiplied.

Kahan’s rearrangement sorts the side lengths and groups the factors to reduce that cancellation. It also allows a failed triangle inequality to be identified before taking a square root, producing zero with an explanation instead of NaN.

Why three angles do not determine size

Triangles with the same three angles are similar, but they may have any positive scale. A triangle with sides 3, 4 and 5 has the same angles as one with sides 6, 8 and 10, while its perimeter and area are different.

The formula, worked line by line

Sides a, b and c lie opposite angles A, B and C. A complete solution uses the selected mode to recover all missing sides and angles before calculating area, perimeter, height, radii and classifications.

The area symbol K is used below to avoid confusing area with angle A. In SSS mode, the sides are additionally sorted as x ≥ y ≥ z before the stable area expression is evaluated.

K = ¼√((x + (y + z))(z − (x − y))(z + (x − y))(x + (y − z)))
A = arccos((b² + c² − a²)/(2bc))
B = arccos((a² + c² − b²)/(2ac))
C = 180° − A − B
a² = b² + c² − 2bc cos A
a/sin A = b/sin B = c/sin C
SSA: c = b cos A ± √(a² − b²sin²A)
Base and height: K = bh/2
P = a + b + c
hₐ = 2K/a
r = K/s, where s = P/2
R = abc/(4K)

Kahan’s area expression is algebraically equivalent to Heron’s formula but arranges the factors to reduce damaging cancellation in needle-shaped triangles. The triangle inequality is checked before its square root is evaluated.

The SSA quadratic can have a negative, zero or positive discriminant. After geometric validation, those cases produce zero, one or as many as two distinct triangle solutions.

Two angles are calculated from side relationships and the third by subtraction from 180°. This makes the displayed angle total exact under the calculator’s rounding convention.

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