What is the Law of Cosines?

The Law of Cosines relates the sides of a triangle to the cosine of one of its angles: c² = a² + b² - 2ab×cos(C). It's useful for solving triangles when you know SSS (three sides) or SAS (two sides and included angle).

What is the Law of Sines?

The Law of Sines states that the ratio of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). It's useful for ASA, AAS, and SSA triangle problems.

How do I find the area of a triangle?

Common formulas: (1) Area = ½ × base × height, (2) Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2, (3) Area = ½ × a × b × sin(C) when you know two sides and included angle.

What is the Pythagorean theorem?

For right triangles only: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle). This is a special case of the Law of Cosines when C = 90°.

Can any three sides form a triangle?

No. The Triangle Inequality states that the sum of any two sides must be greater than the third side. If a + b ≤ c (for any arrangement), no valid triangle exists.

Triangle Calculator

Solve any triangle by entering sides and angles. Calculate area, perimeter, height, angles, and identify triangle type. Uses Law of Sines, Law of Cosines, and Pythagorean theorem with step-by-step solutions.

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Input Values

Known Values

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Decimal Places

2 decimals

4 decimals

6 decimals

Triangle Visualization

Right Triangle

Measurements

Area: 6

Perimeter: 12

Height (h): 4

Angles

∠A: 90°

∠B: 53.13°

∠C: 36.87°

Triangle Type

Right scalene triangle (3-4-5 Pythagorean triple)

Area

Area: 6.0000

Method

Three Sides (SSS)

Area

6.0000

Perimeter

12.0000

Triangle Type

Scalene Right

straighten Sides

Side a

3.0000

Side b

4.0000

Side c

5.0000

change_history Angles

Angle A

36.8699°

Angle B

53.1301°

Angle C

90.0000°

height Heights

h_a

4.0000

h_b

3.0000

h_c

2.4000

Inradius (r)

1.0000

Circumradius (R)

2.5000

functions Step-by-Step Solution

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=== Solving Triangle (SSS) ===

Given: a = 3.0000, b = 4.0000, c = 5.0000

Using Law of Cosines to find angles:

cos(A) = (b² + c² - a²) / (2bc)

cos(A) = (4.0000² + 5.0000² - 3.0000²) / (2 × 4.0000 × 5.0000)

cos(A) = 0.800000

A = 36.8699°

cos(B) = (a² + c² - b²) / (2ac) = 0.600000

B = 53.1301°

C = 180° - A - B = 90.0000°

=== Additional Calculations ===

Perimeter = a + b + c = 12.0000

Semi-perimeter (s) = 6.0000

Area (Heron's formula) = √[s(s-a)(s-b)(s-c)]

Area = 6.0000

Heights (h = 2×Area/side):

hₐ = 4.0000, hᵦ = 3.0000, hᵧ = 2.4000

Inradius (r) = Area/s = 1.0000

Circumradius (R) = abc/(4×Area) = 2.5000

Triangle type: Scalene Right

science Example: Classic 3-4-5 Right Triangle

The 3-4-5 triangle is the most famous Pythagorean triple. With sides a=3, b=4, c=5, it forms a right triangle because 3² + 4² = 9 + 16 = 25 = 5². Area = ½ × 3 × 4 = 6 square units. Perimeter = 3 + 4 + 5 = 12 units.

Expected Results

Area 6

Perimeter 12

Type Right Scalene

How to Use This Calculator

Select Known Values

Choose SSS, SAS, ASA, AAS, SSA, or right triangle.

edit

Enter Measurements

Input the known sides and/or angles.

calculate

Calculate

The calculator solves for all unknown values.

visibility

View Results

See area, perimeter, all sides/angles, and triangle type.

The Formula

Triangles can be solved using Law of Sines, Law of Cosines, or Pythagorean theorem depending on known values.

Area = ½ × base × height = √[s(s-a)(s-b)(s-c)]

lightbulb Variables Explained

s Semi-perimeter: (a+b+c)/2
a, b, c Lengths of the three sides
A, B, C Angles opposite to sides a, b, c

tips_and_updates Pro Tips

The sum of angles in any triangle is always 180°

Triangle inequality: sum of any two sides must be greater than the third side

For right triangles: a² + b² = c² (Pythagorean theorem)

Area = ½ × base × height, or use Heron's formula with three sides

Law of Cosines: c² = a² + b² - 2ab×cos(C) - works for any triangle

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)