Area of Triangle in Terms of Two Sides and Angle
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Corollary to Area of Triangle in Terms of Side and Altitude
The area of a triangle $ABC$ is given by:
- $\dfrac 1 2 a b \sin C$
where:
Proof 1
\(\ds \map \Area {ABC}\) | \(=\) | \(\ds \frac 1 2 h c\) | Area of Triangle in Terms of Side and Altitude | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 h \paren {p + q}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 a b \paren {\frac p a \frac h b + \frac h a \frac q b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 a b \paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta}\) | Definition of Sine of Angle and Definition of Cosine of Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 a b \, \map \sin {\alpha + \beta}\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 a b \sin C\) |
$\blacksquare$
Proof 2
By definition of sine:
- $h = b \sin C$
From Area of Triangle in Terms of Side and Altitude:
- $\map \Area {ABC} = \dfrac {a h} 2$
Substituting:
- $\map \Area {ABC} = \dfrac {a b \sin C} 2$
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(44)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: $4.5$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): triangle (i)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): triangle (i)
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes