Math Calculators ▶ SAS Triangle Calculator
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Give a try to our free SAS triangle calculator to find each and every entity of a triangle. Just provide a couple of adjacent sides of the triangle and their mutual angle and let the calculator do the rest for you.
In Trigonometry, SAS corresponds to a Side-Angle-Side triangle.
What if you think about how to find the angle of a triangle given 2 sides and 1 angle. Let us tell you!
Suppose we have to solve triangle SAS given as under:
In this triangle, we are given:
\(a = 1\)
\(b = 4\)
\(γ = 30^{\text{o}}\)
By using law of cosines:
\(c=\sqrt{\left(1\right)^{2}+\left(4\right)^{2}−2 \text{1*4 } cos\left(30^{\text{o}}\right)}\)
\(c=3.17\) For calculations, tap law of cosine calculator.
For other sides, you may use:
\(a=\sqrt{b^2+c^2−2 \text{ b c } cos(α)}\)
\(b=\sqrt{a^2+c^2−2 \text{ a c } cos(ꞵ)}\)
\(p=a+b+c\)
\(p=1+4+3.17\)
\(p=8.17\)
\(s=\dfrac{p}{2}\)
\(s=\dfrac{8.17}{2}\)
\(s=4.085\)
By using the Heron’s formula:
\(A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\)
\(A=\sqrt{4.085\left(4.085-1\right)\left(4.085-4\right)\left(4.085-3.17\right)}\)
\(A=\sqrt{1}\)
\(A=1\)
\(h_{a}=\dfrac{2A}{a}\)
\(h_{a}=\dfrac{2*1}{1}\)
\(h_{a}=2\)
\(h_{b}=\dfrac{2A}{b}\)
\(h_{b}=\dfrac{2*1}{4}\)
\(h_{b}=\dfrac{1}{2}\)
\(h_{b}=0.5\)
\(h_{c}=\dfrac{2A}{c}\)
\(h_{c}=\dfrac{2*1}{3.17}\)
\(h_{c}=0.630\)
By using law of sines here:
\(\dfrac{b}{sinꞵ}=\dfrac{c}{sin𝛾}\)
Rearranging for missing angle:
\(sinꞵ=\dfrac{b}{c}*sin𝛾\)
\(sinꞵ=\dfrac{4}{3.17}*sin\left(30^{\text{o}}\right)\)
\(sinꞵ=1.261*0.5\)
\(sinꞵ=0.6305\)
\(ꞵ=sin^{-1}\left(0.6305\right)\)
\(ꞵ=140°\)
Now using the supplementary angle measurement
\(𝛼+𝛽+𝛾=180^{\text{o}}\)
\(𝛼+140°56′7″+30°=180^{\text{o}}\)
\(𝛼=180^{\text{o}}-140°-30°\)
\(𝛼=10^{\text{o}}\)
\(r=\dfrac{A}{s}\)
\(r=\dfrac{1}{4.085}\)
\(r=0.244\)
\(R=\dfrac{a*b*c}{4r*s}\)
\(R=\dfrac{1*4*3.17}{4*0.244*4.085}\)
\(R=3.17\)
\(m_{a}=\sqrt{\dfrac{2b^2+2c^2-a^2}{2}}\)
\(m_{a}=\sqrt{\dfrac{2*4^2+23.17^2-1^2}{2}}\)
\(m_{a}=3.576\)
\(m_{b}=\sqrt{\dfrac{2c^2+2a^2-b^2}{2}}\)
\(m_{b}=\sqrt{\dfrac{23.17^2+21^2-4^2}{2}}\)
\(m_{b}=1.239\)
\(m_{c}=\sqrt{\dfrac{2a^2+2b^2-c^2}{2}}\)
\(m_{c}=\sqrt{\dfrac{21^2+24^2-3.17^2}{2}}\)
\(m_{c}=2.446\)
If you wish to use our SAS calculator, read on and understand the following guide!
Input:
Output:
From the source of Wikipedia: Triangle, Types of triangle, Basic facts, Existence of a triangle, Points, lines, and circles associated with a triangle, Computing the sides and angles
From the source of Tutors.com: Triangle Congruence Postulates: SAS, ASA, SSS, AAS, HL
From the source of Lumen Learning: A Bit of Geometry, Similar Triangles