Triangle area with sine: ½·a·b·sin(C)

The formula

For a triangle with two sides $a$ and $b$ and the angle between them $C$:

$\mathcal{A} = \dfrac{1}{2} \times a \times b \times \sin(C)$

Why it works

If $a$ and $b$ come out of vertex $C$, the height relative to $b$ is $a \sin(C)$.

So area = base × height / 2 = $b \times a \sin(C) / 2$.

Example

Triangle with $a = 8$, $b = 5$, included angle $60°$.

$\mathcal{A} = \dfrac{1}{2} \times 8 \times 5 \times \sin(60°) = 20 \times \dfrac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.3$

Alternative with 3 sides (Heron)

If you know the 3 sides $a, b, c$ (no angle):

Semi-perimeter $p = (a + b + c)/2$.

$\mathcal{A} = \sqrt{p(p-a)(p-b)(p-c)}$

Equilateral triangle (side $c$)

Direct formula: $\mathcal{A} = \dfrac{c^2 \sqrt{3}}{4}$.

Check: with sine formula, $\dfrac{1}{2} c \cdot c \sin(60°) = \dfrac{c^2 \sqrt{3}}{4}$. ✓

Reverse application

Given area and 2 sides ⇒ find the angle:

$\sin(C) = \dfrac{2 \mathcal{A}}{a b}$

⚠️ $\sin$ gives 2 possible angles in $[0°, 180°]$ (except 90°).

Common mistakes

1. Using the wrong angle (not the one between the two sides).
2. Forgetting the $1/2$.
3. Sine vs cosine: $\sin$, not $\cos$.
4. Degrees vs radians — check your calculator.

?À toi de jouer

Triangle area: a = 10, b = 6, included angle = 30°?

A.15B.30C.60D.60√3

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