SAS Triangle Area Calculator

What Is the SAS Triangle Area Formula?

The SAS (Side-Angle-Side) formula allows you to calculate the area of any triangle when you know the lengths of two sides and the measure of the angle located directly between them (the included angle).

The standard trigonometric formula for the area of a triangle in SAS form is:

$$\text{Area} = \frac{1}{2} a b \sin(\gamma)$$

Depending on which pair of sides and included angle you have:

• Given sides $a$ and $b$ with included angle $\gamma$: $\text{Area} = \frac{1}{2} a b \sin(\gamma)$
• Given sides $b$ and $c$ with included angle $\alpha$: $\text{Area} = \frac{1}{2} b c \sin(\alpha)$
• Given sides $a$ and $c$ with included angle $\beta$: $\text{Area} = \frac{1}{2} a c \sin(\beta)$

Derivation of the SAS Formula

The basic geometric formula for the area of a triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. Choosing side $a$ as the base, the perpendicular height $h$ from the opposite vertex satisfies:

$$\sin(\gamma) = \frac{h}{b} \implies h = b \sin(\gamma)$$

Substituting $h$ back into the basic area formula yields $\text{Area} = \frac{1}{2} a (b \sin(\gamma)) = \frac{1}{2} ab \sin(\gamma)$.

Solving the Complete Triangle from SAS

Knowing two sides and the included angle uniquely defines the entire triangle. You can determine all remaining parameters:

1. Missing Third Side ($c$): Computed using the Law of Cosines: $$c = \sqrt{a^2 + b^2 - 2ab\cos(\gamma)}$$
2. Remaining Angles ($\alpha, \beta$): Found using the Law of Sines or inverse cosine: $$\alpha = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right), \quad \beta = 180^\circ - \alpha - \gamma$$
3. Perimeter and Altitudes: $P = a + b + c$, with heights to each base given by $h_a = \frac{2 \times \text{Area}}{a}$, $h_b = \frac{2 \times \text{Area}}{b}$, and $h_c = \frac{2 \times \text{Area}}{c}$.

Explore other triangle calculation tools including our 3 Sides Triangle Area Calculator, Area of Triangle with Coordinates Calculator, and Triangle Calculator.