Triangle Calculator

About the Triangle Calculator

This free triangle calculator solves any triangle from three known measurements — sides, angles, or a mix of both. It returns the missing sides and angles along with the area, perimeter, and triangle type, computed locally in your browser using the Law of Sines, the Law of Cosines, and Heron's formula.

What is a Triangle Calculator?

A triangle calculator solves for the unknown sides and angles of a triangle when you supply enough information to define it. Three measurements are generally sufficient, provided at least one is a side — the three angles alone leave the triangle's size undetermined.

How to Use

Select the tab that matches the data you have, enter the known values, and click Calculate. Angles are entered in degrees.

1. SSS: enter all three side lengths.
2. SAS: enter two sides and the angle between them.
3. ASA: enter two angles and the side between them.
4. AAS: enter two angles and any one side.
5. SSA: enter two sides and a non-included angle — the ambiguous case, where two solutions may exist.

Essential Triangle Formulas

Perimeter

P = a + b + c

• a, b, c — the three side lengths

The perimeter is the total distance around the triangle. For sides of 3, 4, and 5, P = 3 + 4 + 5 = 12 units.

Area from Base and Height

Area = ½ × base × height

• base — any side of the triangle
• height — the perpendicular distance from that base to the opposite vertex

Use it whenever the height is known — it also powers our area calculator. When only the three sides are known, Heron's formula below is the practical choice.

Heron's Formula (Three Sides Only)

s = (a + b + c) / 2

Area = √(s(s − a)(s − b)(s − c))

• s — the semi-perimeter (half the perimeter)
• a, b, c — the three side lengths

Worked example — the 3-4-5 triangle. With sides a = 3, b = 4, c = 5:

1. Semi-perimeter: s = (3 + 4 + 5) / 2 = 6
2. Substitute: Area = √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = √(6 × 3 × 2 × 1) = √36
3. Area = 6 square units

Cross-check with ½ × base × height: ½ × 4 × 3 = 6, the same answer — the 3-4-5 triangle is right-angled, so its legs serve as base and height. The SSS tab above confirms this: area 6.0000, perimeter 12.0000, angles 36.87°, 53.13°, 90.00°.

Angle Sum

A + B + C = 180°

The three interior angles of every triangle total exactly 180 degrees, so two known angles always reveal the third. Worked example: if two angles measure 90° and 35°, the third is 180° − 90° − 35° = 55°.

Types of Triangles

Triangles are classified independently by side lengths and by angles; every triangle gets one label from each group — the 3-4-5 triangle is both scalene and right.

The Triangle Inequality Rule

Not every set of three lengths can form a triangle. The triangle inequality theorem states that the sum of any two sides must be greater than the third side:

a + b > c  a + c > b  b + c > a

Valid: 3, 4, and 5 work because 3 + 4 = 7 > 5, 3 + 5 = 8 > 4, and 4 + 5 = 9 > 3. Invalid: 2, 3, and 6 fail because 2 + 3 = 5, which is less than 6 — the two short sides cannot meet. The SSS tab alerts you when this rule is violated.

Law of Sines and Law of Cosines

Right triangles yield to the Pythagorean theorem — try our Pythagorean theorem calculator — but non-right triangles need two more general tools. The Law of Sines relates each side to the sine of its opposite angle:

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

It solves ASA, AAS, and SSA cases — for example, with A = 30°, B = 70°, and included side c = 10, the ASA tab finds C = 80°, a ≈ 5.0771, and b ≈ 9.5419. The Law of Cosines generalizes the Pythagorean theorem to any angle:

c² = a² + b² − 2ab × cos(C)

It solves SSS and SAS cases; when C = 90°, cos(C) = 0 and the formula collapses to the familiar c² = a² + b². For shapes beyond triangles, see the area calculator.

Frequently Asked Questions