ASA Triangle Calculator

What Is an ASA Triangle?

In Euclidean geometry, ASA (Angle-Side-Angle) refers to a triangle where two interior angles and the length of the side connected directly between them (the included side) are known.

According to the ASA Congruence Postulate, specifying two angles and their included side uniquely determines the size and shape of the entire triangle.

How to Solve an ASA Triangle

Solving an ASA triangle involves a direct three-step procedure:

1. Find the Third Angle ($\gamma$): Because the interior angles of any planar triangle sum to $180^\circ$: $$\gamma = 180^\circ - \alpha - \beta$$
2. Find the Unknown Sides ($a, b$) Using the Law of Sines: $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$ $$a = \frac{c \sin(\alpha)}{\sin(\gamma)}, \quad b = \frac{c \sin(\beta)}{\sin(\gamma)}$$
3. Calculate the Area: Using the trigonometric area formula: $$\text{Area} = \frac{1}{2} a b \sin(\gamma) = \frac{c^2 \sin(\alpha) \sin(\beta)}{2 \sin(\alpha + \beta)}$$

Worked Example of an ASA Triangle

Suppose a triangle has angles $\alpha = 50^\circ$, $\beta = 70^\circ$, and included side $c = 8$:

• Third angle: $\gamma = 180^\circ - 50^\circ - 70^\circ = 60^\circ$.
• Side $a = \frac{8 \times \sin(50^\circ)}{\sin(60^\circ)} = \frac{8 \times 0.76604}{0.86603} \approx 7.0764$.
• Side $b = \frac{8 \times \sin(70^\circ)}{\sin(60^\circ)} = \frac{8 \times 0.93969}{0.86603} \approx 8.6805$.
• Perimeter: $P = 7.0764 + 8.6805 + 8 = 23.7569$.
• Area: $\text{Area} = 0.5 \times 7.0764 \times 8.6805 \times \sin(60^\circ) \approx 26.5986\text{ sq units}$.

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