Law of Cosines Calculator

About Law of Cosines Calculator

Solve triangles using the law of cosines: c² = a² + b² - 2ab cos(C). Find a missing side from two sides and the included angle (SAS), or find all three angles from three known sides (SSS). Step-by-step working with area and perimeter included.

The Law of Cosines Formula

The law of cosines relates the sides and angles of any triangle:

c² = a² + b² - 2ab cos(C)

There are three forms, one for each side:

To Find	Formula
Side c	c² = a² + b² - 2ab cos(C)
Side b	b² = a² + c² - 2ac cos(B)
Side a	a² = b² + c² - 2bc cos(A)

To find an angle when all three sides are known, rearrange:

cos(C) = (a² + b² - c²) / (2ab)

Worked Example: SAS (Find Missing Side)

Given: a = 7, b = 10, C = 55°

1. c² = 7² + 10² - 2(7)(10)cos(55°)
2. c² = 49 + 100 - 140 × 0.5736
3. c² = 149 - 80.30 = 68.70
4. c = √68.70 = 8.289

Now find the remaining angles using the rearranged formula:

5. cos(A) = (b² + c² - a²) / (2bc) = (100 + 68.70 - 49) / (2 × 10 × 8.289) = 119.70/165.78 = 0.7222
6. A = arccos(0.7222) = 43.73°
7. B = 180° - 55° - 43.73° = 81.27°

Area: (1/2) × 7 × 10 × sin(55°) = 35 × 0.8192 = 28.67 square units

Worked Example: SSS (Find All Angles)

Given: a = 5, b = 7, c = 9

1. First check triangle inequality: 5+7=12 > 9, 5+9=14 > 7, 7+9=16 > 5 ✓
2. Find largest angle first (opposite longest side c):
3. cos(C) = (5² + 7² - 9²) / (2 × 5 × 7) = (25 + 49 - 81) / 70 = -7/70 = -0.1
4. C = arccos(-0.1) = 95.74° (obtuse - as expected for the longest side)
5. cos(A) = (7² + 9² - 5²) / (2 × 7 × 9) = (49 + 81 - 25) / 126 = 105/126 = 0.8333
6. A = arccos(0.8333) = 33.56°
7. B = 180° - 95.74° - 33.56° = 50.70°

Verify: 33.56° + 50.70° + 95.74° = 180.00° ✓

Connection to the Pythagorean Theorem

When angle C = 90°, cos(90°) = 0, so the law of cosines becomes:

c² = a² + b² - 2ab(0) = a² + b²

This is exactly the Pythagorean theorem. The law of cosines is the general version that works for all triangles, not just right triangles.

Angle C	cos(C)	Effect on c²	Triangle Type
Less than 90°	Positive	c² < a² + b²	Acute
Exactly 90°	0	c² = a² + b²	Right
Greater than 90°	Negative	c² > a² + b²	Obtuse

When to Use Law of Cosines vs Law of Sines

Given Information	Best Method	Why
SAS (two sides + included angle)	Law of Cosines	Directly finds the third side
SSS (three sides)	Law of Cosines	Directly finds any angle
AAS (two angles + any side)	Law of Sines	Simpler calculation
ASA (two angles + included side)	Law of Sines	Find third angle first, then use ratios
SSA (two sides + non-included angle)	Law of Sines	But watch for ambiguous case

Triangle Area from Three Sides (Heron's Formula)

When you know all three sides, you can find the area without computing any angles:

s = (a + b + c) / 2 (semi-perimeter)

Area = √(s(s-a)(s-b)(s-c))

Example: a = 5, b = 7, c = 9

• s = (5 + 7 + 9) / 2 = 10.5
• Area = √(10.5 × 5.5 × 3.5 × 1.5) = √303.1875 = 17.41 square units

The Triangle Inequality

Before solving, the calculator checks that a valid triangle can exist. The triangle inequality states that the sum of any two sides must be greater than the third side:

• a + b > c
• a + c > b
• b + c > a

If any of these fail, no triangle exists and the calculator shows an error.

For SSA problems with possible ambiguous solutions, the law of sines calculator checks for multiple triangles. For right triangle problems specifically, the Pythagorean theorem calculator provides a simpler interface.

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