Triangle calculator SSA

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

Sides: a = 34   b = 46   c = 38.8499034075

Area: T = 646.0021521012
Perimeter: p = 118.8499034075
Semiperimeter: s = 59.42545170375

Angle ∠ A = α = 46.3010955004° = 46°18'3″ = 0.80881041116 rad
Angle ∠ B = β = 78° = 1.36113568166 rad
Angle ∠ C = γ = 55.6999044996° = 55°41'57″ = 0.97221317254 rad

Height: ha = 388.0000894713
Height: hb = 28.08770226527
Height: hc = 33.25770184249

Median: ma = 39.03436229971
Median: mb = 28.34882578703
Median: mc = 35.47879951218

Inradius: r = 10.87109595503
Circumradius: R = 23.51438336819

Vertex coordinates: A[38.8499034075; 0] B[0; 0] C[7.06989974878; 33.25770184249]
Centroid: CG[15.30660105209; 11.08656728083]
Coordinates of the circumscribed circle: U[19.42545170375; 13.25109815591]
Coordinates of the inscribed circle: I[13.42545170375; 10.87109595503]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.6999044996° = 133°41'57″ = 0.80881041116 rad
∠ B' = β' = 102° = 1.36113568166 rad
∠ C' = γ' = 124.3010955004° = 124°18'3″ = 0.97221317254 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 34 ; ; b = 46 ; ; beta = 78° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 46**2 = 34**2 + c**2 -2 * 46 * c * cos (78° ) ; ; ; ; c**2 -14.138c -960 =0 ; ; p=1; q=-14.1379949756; r=-960 ; ; D = q**2 - 4pr = 14.138**2 - 4 * 1 * (-960) = 4039.88290193 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 14.14 ± sqrt{ 4039.88 } }{ 2 } ; ; c_{1,2} = 7.0689974878 ± 31.7800365872 ; ; c_{1} = 38.849034075 ; ;
c_{2} = -24.7110390994 ; ; ; ; (c -38.849034075) (c +24.7110390994) = 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 = 34 ; ; b = 46 ; ; c = 38.85 ; ;

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

p = a+b+c = 34+46+38.85 = 118.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.85 }{ 2 } = 59.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.42 * (59.42-34)(59.42-46)(59.42-38.85) } ; ; T = sqrt{ 417317.97 } = 646 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 646 }{ 34 } = 38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 646 }{ 46 } = 28.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 646 }{ 38.85 } = 33.26 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 34**2-46**2-38.85**2 }{ 2 * 46 * 38.85 } ) = 46° 18'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 46**2-34**2-38.85**2 }{ 2 * 34 * 38.85 } ) = 78° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.85**2-34**2-46**2 }{ 2 * 46 * 34 } ) = 55° 41'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 646 }{ 59.42 } = 10.87 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 34 }{ 2 * sin 46° 18'3" } = 23.51 ; ;




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