Triangle area from two sides: use the angle between them
Apply one-half ab sin C and distinguish an included angle from another angle in the triangle.
Method
Two side lengths and their included angle determine a triangle’s area. The included angle is the angle where those two sides meet. The formula is one-half multiplied by the two lengths and the sine of that angle. An angle at another corner cannot simply be inserted with the same pair of sides.
The formula follows from the base-times-height rule. If one side is the base a, the component of side b perpendicular to it has height b sin C. Multiplying base by height and dividing by two gives the area. Both lengths must use the same unit, and the result uses that unit squared. Convert centimetres to metres before calculating if you want square metres.
- InputsBase · Perpendicular height · Side a
- MethodArea = base*height/2; Perimeter = a+b+c
- ResultArea · Perimeter
Example only; replace these assumptions with your own values.
Worked example
For sides 8 cm and 5 cm with an included angle of 30 degrees, the area is 10 square centimetres. With the same side lengths and an included angle of 90 degrees, the area is 20 square centimetres. An included angle of 150 degrees also gives 10 square centimetres because its sine equals the sine of 30 degrees. Equal area does not imply the same triangle shape.
Use the Scientific calculator in degree mode when entering degree angles. In radian mode, 30 degrees must first be converted to pi divided by 6 radians. For the base-height Triangle geometry calculator, enter a base of 8 and the computed perpendicular height of 2.5 for the 30-degree example. That produces the same area of 10.
A = 0.5 × a × b × sin(C)
0.5 × 8 × 5 × sin(30°) = 10 cm²
h = 5 × sin(30°) = 2.5 cm
0.5 × 8 × 2.5 = 10 cm²
- Included angle 30°
- 10 cm²
- Included angle 90°
- 20 cm²
- Included angle 150°
- 10 cm²
Area = 0.5 × 8 × 5 × sin(included angle).
Checks and limits
A non-degenerate triangle requires positive side lengths and an included angle strictly between 0 and 180 degrees. At the endpoints the height and area are zero. Do not substitute the sloping side itself for the perpendicular height unless the angle is 90 degrees. The calculation describes ideal geometry; measurements with rounding or uncertainty give an estimated area.
Related calculators
Source: OpenStax