Library / Foundations: Maths / FMA-16
Foundations: Maths
Triangle area from two sides and an angle
Area when two sides and the included angle are known.
- Ref
- FMA-16
- Rev
- v1
- Field
- Foundations: Maths
- Result
- A [m²]
a and b are two sides and C the angle between them. The calculator covers C up to 90°; angles above 90° give the same area as 180° − C.
The area is half the product of two sides, scaled by how wide the angle between them is.
Calculator
m²
m
m
°
- Maximum possible area (C = 90°)20 m²
- Height on side a2.5 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Area (m²) current value 10 m²
bSide bKey driver
2200
aSide a
1.25125
CIncluded angle
3.4720
- At 90° the formula reduces to ½ × base × height.
- Area is symmetric: an angle of 150° gives the same area as 30°.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Side b | 1 to 100 m | 2 | 200 | 198 | +10% | +10 % |
| a Side a | 1 to 100 m | 1.25 | 125 | 123.7 | +10% | +10 % |
| C Included angle | 10 to 90 ° | 3.473 | 20 | 16.53 | +10% | +8.93 % |
Sweep of Included angle
| Included angle (°) | Area (m²) |
|---|---|
| 10 | 3.473 |
| 20 | 6.84 |
| 30 | 10 |
| 40 | 12.86 |
| 50 | 15.32 |
| 60 | 17.32 |
| 70 | 18.79 |
| 80 | 19.7 |
| 90 | 20 |
Sweep one variable
Area (m²)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
A
Area
What the equation gives you. Everything else feeds it.
a
Side a
One side. Directly proportional.
b
Side b
The other side. Directly proportional.
C
Included angle
Angle between the sides. Area is largest at 90°.
Worked example with the values above
- sin 30° = 0.5.
- ½ab = 0.5 × 8 × 5 = 20.
- Area = 10 m².
Assumptions and limits
- C lies between sides a and b.
- Angle in degrees.
Common mistakes
- Using an angle that is not between the two sides.
- Calculator in radian mode.
Statistics and uncertainty
At small angles the area is proportional to the angle, so a 1° error on a 10° angle gives a 10% error in area.
Where it is used
- Land and site areas from surveyed sides.
- Coverage area between two bearings.
Maths you need
- Trigonometry: sin of an angle
- Triangle area: ½ × base × height