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Use our free SAS triangle calculator to compute all properties of a triangle. Just provide two adjacent sides and the included angle, and the calculator will automatically find the rest.
In trigonometry, SAS stands for Side-Angle-Side—a triangle where two sides and the included angle are known. This is one of the key triangle congruence postulates.
Consider a triangle with:
\( a = 1, \quad b = 4, \quad \gamma = 30^\circ \)
Step 1: Find the third side \(c\) using the Law of Cosines:
\( c = \sqrt{a^2 + b^2 - 2ab \cos \gamma} \)
\( c = \sqrt{1^2 + 4^2 - 2 \cdot 1 \cdot 4 \cdot \cos 30^\circ} \approx 3.17 \)
Step 2: Calculate the perimeter:
\( p = a + b + c = 1 + 4 + 3.17 \approx 8.17 \)
Step 3: Calculate the semiperimeter:
\( s = \frac{p}{2} = \frac{8.17}{2} \approx 4.085 \)
Step 4: Compute the area using Heron's formula:
\( A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{4.085(4.085-1)(4.085-4)(4.085-3.17)} \approx 1 \)
Step 5: Find the heights:
\( h_a = \frac{2A}{a} = \frac{2 \cdot 1}{1} = 2 \)
\( h_b = \frac{2A}{b} = \frac{2 \cdot 1}{4} = 0.5 \)
\( h_c = \frac{2A}{c} = \frac{2 \cdot 1}{3.17} \approx 0.63 \)
Step 6: Find the remaining angles using the Law of Sines:
\( \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \)
\( \sin \beta = \frac{b}{c} \sin \gamma = \frac{4}{3.17} \cdot \sin 30^\circ \approx 0.63 \)
\( \beta \approx 39.1^\circ, \quad \alpha = 180^\circ - \beta - \gamma \approx 110.9^\circ \)
Step 7: Calculate the inradius and circumradius:
\( r = \frac{A}{s} = \frac{1}{4.085} \approx 0.244 \)
\( R = \frac{abc}{4A} = \frac{1 \cdot 4 \cdot 3.17}{4 \cdot 1} \approx 3.17 \)
Step 8: Calculate the medians:
\( m_a = \sqrt{\frac{2b^2 + 2c^2 - a^2}{2}} \approx 3.576 \)
\( m_b = \sqrt{\frac{2c^2 + 2a^2 - b^2}{2}} \approx 1.239 \)
\( m_c = \sqrt{\frac{2a^2 + 2b^2 - c^2}{2}} \approx 2.446 \)
Our calculator makes solving SAS triangles fast and easy:
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Wikipedia: Triangle Basics
Tutors.com: SAS, ASA, SSS Postulates
Lumen Learning: Similar Triangles & Geometry
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