Triangle Calculator | TheSmartCalculator

Side A

Side B

Side C

About the Triangle Calculator

Triangles are the fundamental building block of geometry — and solving an unknown triangle is one of the most common tasks in trigonometry, physics, navigation, engineering, and construction. Our Triangle Calculator solves any triangle when you provide at least three pieces of information (sides and/or angles, with at least one being a side).

Whether you need the Pythagorean theorem for a right triangle or the Law of Sines and Cosines for an oblique triangle, this tool handles it automatically — just enter what you know and it calculates the rest.

How It Works

Enter any combination of three known values (sides a, b, c and angles A, B, C). The calculator detects the configuration (SSS, SAS, ASA, AAS, or SSA) and applies the appropriate theorem or formula to solve for the remaining unknowns. It also computes the area, perimeter, and classifies the triangle by sides and angles.

Formula / Key Reference

Pythagorean Theorem (right triangles): c² = a² + b²

Law of Cosines (any triangle):

c² = a² + b² − 2ab·cos(C)

Law of Sines:

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

Area (Heron's formula):

Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2

Real-World Example

Construction scenario: A roof forms a triangle. You know the base (run) is 24 feet, the rise is 8 feet, and the angle at the peak is unknown.

This is a right triangle (the vertical rise meets the horizontal run at 90°):

Hypotenuse (rafter) = √(24² + 8²) = √(576 + 64) = √640 = 25.3 feet

Pitch angle at the base = arctan(8 ÷ 24) = arctan(0.333) = 18.4°

Angle at the peak = 90° − 18.4° = 71.6°

Check: 18.4° + 71.6° + 90° = 180° ✓

Area = (1/2) × base × height = (1/2) × 24 × 8 = 96 square feet

Perimeter = 24 + 8 + 25.3 = 57.3 feet

The contractor now knows to order at least 25.5 feet of rafter material per roof section.

Common Uses

Architecture and construction: calculating roof pitch, rafter lengths, and structural angles

Navigation: solving triangulation problems for position finding

Physics: resolving force vectors into components

Surveying: calculating distances and angles from measurement data

Trigonometry homework and standardized test preparation (SAT, ACT)

Frequently Asked Questions

What is the SSA ambiguous case? ▼

When you know two sides and an angle not between them (SSA), there may be 0, 1, or 2 valid triangles depending on the given values. This is called the ambiguous case. The calculator detects this and presents all valid solutions. Check which solution fits your physical context.

When should I use the Law of Sines vs. Law of Cosines? ▼

Use the Law of Sines when you know an angle and its opposite side (AAS or ASA configurations). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle (SAS). For right triangles, the Pythagorean theorem is simpler and more direct.

What is the sum of angles in a triangle always equal to? ▼

The interior angles of any triangle in Euclidean (flat) geometry always sum to exactly 180°. This is true regardless of the triangle's shape or size. If you know two angles, the third is always 180° minus the sum of the other two. Our calculator verifies this as a built-in check.

What is Heron****'****s formula and when is it useful? ▼

Heron's formula calculates the area of a triangle using only the three side lengths, without needing the height. It is named after Hero of Alexandria (around 60 AD). It is most useful in surveying, construction, and any situation where you know the sides but not the perpendicular height of the triangle.