Law of Cosines Calculator
The third side from two sides and the angle between them, or any angle from the three sides, with the substitution shown.
Results
The one that was missing
6.2450
Side c
6.2450
m
Angle C, opposite side c
60.0000
°
The substitution, term by term
25 + 49 - 35
Angle A, opposite side a
43.8979
°
Angle B, opposite side b
76.1021
°
Area
15.1554
m²
Perimeter
18.2450
m
Is this just Pythagoras?
No
By its angles
Acute — every angle under 90°
Side · its length · angle · its size
a 5 A 43.897886
b 7 B 76.102114
c 6.244998 C 60
What this tool does
The cosine rule is Pythagoras with a correction term for the angle. When that angle is a right angle its cosine is zero, the correction disappears, and what is left is a² + b². The page points that out when it happens, because seeing the two rules as one is worth more than memorising both.
Formula
c² = a² + b² − 2ab cos C · cos C = (a² + b² − c²) ÷ 2ab
Variables
| Symbol | Meaning | Unit |
|---|---|---|
find | What you are looking for | — |
a | Side a | m |
b | Side b | m |
gamma | The angle between the two sides | ° |
c | Side c | m |
OU | The one that was missing | — |
TS | Side c | m |
TA | Angle C, opposite side c | ° |
ST | The substitution, term by term | — |
AA | Angle A, opposite side a | ° |
AB | Angle B, opposite side b | ° |
AR | Area | m² |
PE | Perimeter | m |
PY | Is this just Pythagoras? | — |
KD | By its angles | — |
TB | Side · its length · angle · its size | — |
Worked example
- What you are looking forside
- Side a5 m
- Side b7 m
- The angle between the two sides60 °
- Side c6 m
- The one that was missing6.2450
- Side c6.2450 m
- Angle C, opposite side c60.0000 °
- The substitution, term by term25 + 49 - 35
- Angle A, opposite side a43.8979 °
- Angle B, opposite side b76.1021 °
- Area15.1554 m²
- Perimeter18.2450 m
- Is this just Pythagoras?No
- By its anglesAcute — every angle under 90°
- Side · its length · angle · its sizea 5 A 43.897886 b 7 B 76.102114 c 6.244998 C 60
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.