Math101 · Independent Ontario learning library
Math101Area of an Oblique Triangle
An oblique triangle has no right angle. Its area can be found from two sides and their included angle: $A=\frac12ab\sin C$, where $C$ lies between sides $a$ and $b$.
The sine-area formula handles surveying, navigation, and design when altitude is not directly known. It also leads to Heron-type and law-of-sines area relationships.
Intuition and core definition
An oblique triangle has no right angle. Its area can be found from two sides and their included angle: $A=\frac12ab\sin C$, where $C$ lies between sides $a$ and $b$. Equivalent forms cycle the labels. The formula comes from height $h=b\sin C$ in the familiar $A=\frac12(\text{base})(\text{height})$.
Notation, language, and conditions
Standard notation places side $a$ opposite angle $A$, and similarly for $b,c$. The included angle is formed by the two given sides. Angles must match the calculator’s degree or radian mode. Since area is nonnegative, an ordinary triangle uses $0^\circ<C<180^\circ$ and $\sin C>0$.
Why this idea matters
The formula $A=\frac12 ab\sin C$ turns two sides and their included angle into area by extracting the perpendicular height.
A dependable method
- Sketch and label the two known sides and their included angle.
- Verify that the angle is between those sides.
- Choose the matching form $\frac12ab\sin C$.
- Set calculator angle mode, substitute, and preserve guard digits.
- Report square units and check against $ab/2$, the maximum for those sides.
