Triangle Calculator

Enter SSS, SAS, ASA, AAS, or a right triangle. You get every side, angle, and the area.

Case *

Side a *

Side b *

Side c *

SSA is not solved on this page. Two sides and a non-included angle can make two triangles, one triangle, or none.

Smart Tips

• Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

• A right triangle is a mode here. The right angle is C, so c is the hypotenuse.

• SSA is skipped on purpose. Use SSS, SAS, ASA, AAS, or the right-triangle mode.

• Sides that fail the triangle inequality, or angles that do not sum to 180°, are rejected.

The Formula

c² = a² + b² − 2ab cos C

SSS = Law of cosines
Three sides give every angle

SAS = Law of cosines
Two sides and the included angle

ASA = Law of sines
Two angles and the included side

AAS = Law of sines
Two angles and a non-included side

Area uses Heron’s formula after the three sides are known: s = (a + b + c) / 2 and area = √(s(s − a)(s − b)(s − c)). That is more than base times height. In a right triangle, cos 90° = 0, so the law of cosines reduces to a² + b² = c².

Key Insights

Unique cases

SSS, SAS, ASA, and AAS each determine one triangle when the numbers are possible. The page rejects a flat or broken triangle.

Ambiguous SSA

Two sides and an angle opposite one of them can fit two triangles. This page does not guess which one you mean.

Right triangle mode

Enter both legs, or a leg and the hypotenuse. The right angle stays at C. Other oblique cases stay on this same URL.

Area is not base times height

The area calculator’s triangle is base times height. This page returns the missing sides and angles, then the area from the three sides.

Frequently Asked Questions

SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (angle A, angle B, and side a opposite angle A), and a right triangle from both legs or from a leg and the hypotenuse. The right angle in that mode is C.

SSA means two sides and an angle that is not between them. That is the ambiguous case: the numbers can fit two triangles, one triangle, or none. This page does not solve SSA. Use SAS when the angle is the included angle.

Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. In SAS, angle A sits between sides b and c. In ASA, side c sits between angles A and B. In AAS, side a is opposite angle A, so it is not the included side.

The sum of any two sides must be greater than the third side. Every angle must be greater than 0° and less than 180°, and the three angles must sum to 180°. A hypotenuse that is not longer than a leg is rejected. Division by zero in the law of sines is also rejected.

The area calculator’s triangle uses base times height only. This page solves the triangle first, then finds the area with Heron’s formula from the three sides. The Pythagorean theorem calculator is a separate page for a² + b² = c² only.

SSS and SAS use the Law of Cosines from OpenStax Algebra and Trigonometry 2e, section 10.2. ASA and AAS use the Law of Sines from section 10.1 of the same book. Section 10.1 calls SSA the ambiguous case, which is why this page skips it. Area is Heron’s formula from the three sides. No outside data feed is used.

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Area

Enter values to calculate

Side a

Side b

Side c

Angle A

Angle B

Angle C

Note:Angles are in degrees. SSA is not solved, because two sides and a non-included angle can be ambiguous. Area uses Heron’s formula after the sides are known. A zero side or a zero angle is rejected.