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Triangle Calculator

Solve right triangles (enter any 2 sides) or any triangle using SSS (3 sides) or SAS (2 sides + included angle). Calculates all sides, all angles, area, and perimeter using the Pythagorean theorem, law of sines, and law of cosines.

Right Triangle (a² + b² = c²)

Enter any 2 sides. Leave the third blank. c = hypotenuse.

Leg a:3
Leg b:4
Hypotenuse c:5
Angle A:36.8699°
Angle B:53.1301°
Angle C:90°
Area:6
Perimeter:12
Given: a = 3, b = 4

Using Pythagorean theorem: c = √(a² + b²)
  c = √(3² + 4²)
  c = √(9 + 16)
  c = √25
  c = 5

Angles:
  A = arcsin(a/c) = arcsin(3/5) = 36.8699°
  B = arcsin(b/c) = arcsin(4/5) = 53.1301°
  C = 90°

Right Triangle Trigonometry

A right triangle has one 90° angle. The side opposite the right angle is the hypotenuse (c), the longest side. The other two sides are legs (a and b).

Pythagorean theorem: a² + b² = c²

The trigonometric functions relate angles to side ratios. For angle A (opposite side a, adjacent side b, hypotenuse c):

Function Formula Mnemonic
sin(A) opposite / hypotenuse = a/c SOH
cos(A) adjacent / hypotenuse = b/c CAH
tan(A) opposite / adjacent = a/b TOA

The mnemonic SOH-CAH-TOA helps remember these definitions. Inverse functions (arcsin, arccos, arctan) convert a ratio back to an angle.

Special Right Triangles

Type Angles Side ratio Example
45-45-90 45°, 45°, 90° 1 : 1 : √2 legs 5, 5 → hyp 5√2 ≈ 7.071
30-60-90 30°, 60°, 90° 1 : √3 : 2 short leg 3, long leg 3√3 ≈ 5.196, hyp 6
3-4-5 37°, 53°, 90° 3 : 4 : 5 3² + 4² = 9 + 16 = 25 = 5²

The Law of Sines

For any triangle with sides a, b, c opposite angles A, B, C:

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

The law of sines is used when you know:

The Law of Cosines

The law of cosines generalizes the Pythagorean theorem to non-right triangles:

c² = a² + b² − 2ab × cos(C)

Similarly: a² = b² + c² − 2bc × cos(A), and b² = a² + c² − 2ac × cos(B).

Use the law of cosines when you know:

Heron's Formula for Area

When all three sides are known, compute area without needing a height using Heron's formula:

  1. Compute semi-perimeter: s = (a + b + c) / 2
  2. Area = √(s × (s−a) × (s−b) × (s−c))

Example: Triangle with sides 5, 6, 7: s = 9, Area = √(9×4×3×2) = √216 ≈ 14.70.

Frequently Asked Questions

What is the law of sines?

The law of sines states: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides opposite angles A, B, C respectively. It is used to solve triangles when you know two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA). The ratio of any side to the sine of its opposite angle is constant for a given triangle.

What is the law of cosines?

The law of cosines: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b. It generalizes the Pythagorean theorem — when C = 90°, cos(90°) = 0, and the formula reduces to c² = a² + b². Use it for SSS (find angles from sides) or SAS (find the third side from two sides and the included angle).

How do you find the area of a triangle?

Several formulas exist: (1) Base-height: A = ½ × b × h, where h is perpendicular height. (2) Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. (3) SAS formula: A = ½ab·sin(C) where C is the angle between sides a and b. (4) Coordinate formula: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| for vertices (x₁,y₁), (x₂,y₂), (x₃,y₃).

What are the triangle inequality rules?

For three lengths to form a valid triangle, the sum of any two sides must be greater than the third: a + b > c, a + c > b, and b + c > a. If any condition fails, no triangle exists. A degenerate case occurs when one side equals the sum of the other two (a + b = c), creating a "flat triangle" with zero area. For right triangles, additionally c must be greater than both a and b.

What are the types of triangles?

By sides: Equilateral (all sides equal, all angles 60°), Isosceles (two equal sides, two equal angles), Scalene (all sides different). By angles: Acute (all angles < 90°), Right (one angle exactly 90°), Obtuse (one angle > 90°). Special right triangles: 45-45-90 (legs equal, hypotenuse = leg × √2) and 30-60-90 (sides in ratio 1:√3:2).

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