Triangle calculator SSA

Please enter two sides and a non-included angle
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Obtuse scalene triangle.

Sides: a = 3.5   b = 4.2   c = 2.26113573592

Area: T = 3.95767726511
Perimeter: p = 9.96113573592
Semiperimeter: s = 4.98106786796

Angle ∠ A = α = 56.43295369338° = 56°25'46″ = 0.98548812149 rad
Angle ∠ B = β = 91° = 1.58882496193 rad
Angle ∠ C = γ = 32.57704630662° = 32°34'14″ = 0.56884618194 rad

Height: ha = 2.26110129435
Height: hb = 1.88441774529
Height: hc = 3.4999466933

Median: ma = 2.88334646786
Median: mb = 2.06768499106
Median: mc = 3.69768318495

Inradius: r = 0.79444243959
Circumradius: R = 2.11003198889

Vertex coordinates: A[2.26113573592; 0] B[0; 0] C[-0.06110834225; 3.4999466933]
Centroid: CG[0.73334246455; 1.16664889777]
Coordinates of the circumscribed circle: U[1.13106786796; 1.77700026438]
Coordinates of the inscribed circle: I[0.78106786796; 0.79444243959]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.5770463066° = 123°34'14″ = 0.98548812149 rad
∠ B' = β' = 89° = 1.58882496193 rad
∠ C' = γ' = 147.4329536934° = 147°25'46″ = 0.56884618194 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.5 ; ; b = 4.2 ; ; beta = 91° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.2**2 = 3.5**2 + c**2 -2 * 3.5 * c * cos (91° ) ; ; ; ; c**2 +0.122c -5.39 =0 ; ; p=1; q=0.122; r=-5.39 ; ; D = q**2 - 4pr = 0.122**2 - 4 * 1 * (-5.39) = 21.574924738 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ -0.12 ± sqrt{ 21.57 } }{ 2 } ; ; c_{1,2} = -0.06108342 ± 2.3224407817 ; ; c_{1} = 2.2613573617 ; ;
c_{2} = -2.3835242017 ; ; ; ; text{ Factored form: } ; ; (c -2.2613573617) (c +2.3835242017) = 0 ; ; ; ; c>0 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 3.5 ; ; b = 4.2 ; ; c = 2.26 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 3.5+4.2+2.26 = 9.96 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.96 }{ 2 } = 4.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.98 * (4.98-3.5)(4.98-4.2)(4.98-2.26) } ; ; T = sqrt{ 15.66 } = 3.96 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.96 }{ 3.5 } = 2.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.96 }{ 4.2 } = 1.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.96 }{ 2.26 } = 3.5 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 4.2**2+2.26**2-3.5**2 }{ 2 * 4.2 * 2.26 } ) = 56° 25'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.5**2+2.26**2-4.2**2 }{ 2 * 3.5 * 2.26 } ) = 91° ; ; gamma = 180° - alpha - beta = 180° - 56° 25'46" - 91° = 32° 34'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.96 }{ 4.98 } = 0.79 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.5 }{ 2 * sin 56° 25'46" } = 2.1 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.2**2+2 * 2.26**2 - 3.5**2 } }{ 2 } = 2.883 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.26**2+2 * 3.5**2 - 4.2**2 } }{ 2 } = 2.067 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.2**2+2 * 3.5**2 - 2.26**2 } }{ 2 } = 3.697 ; ;
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