Triangle calculator SSA

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

Sides: a = 50   b = 90   c = 129.7599352518

Area: T = 1621.992190648
Perimeter: p = 269.7599352518
Semiperimeter: s = 134.8879676259

Angle ∠ A = α = 16.12876202132° = 16°7'39″ = 0.28114800732 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 133.8722379787° = 133°52'21″ = 2.33765138048 rad

Height: ha = 64.88796762591
Height: hb = 36.04442645884
Height: hc = 25

Median: ma = 108.8298970329
Median: mb = 87.42985124142
Median: mc = 33.02546515275

Inradius: r = 12.02554730102
Circumradius: R = 90

Vertex coordinates: A[129.7599352518; 0] B[0; 0] C[43.30112701892; 25]
Centroid: CG[57.68768742358; 8.33333333333]
Coordinates of the circumscribed circle: U[64.88796762591; -62.37548956594]
Coordinates of the inscribed circle: I[44.88796762591; 12.02554730102]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.8722379787° = 163°52'21″ = 0.28114800732 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 46.12876202132° = 46°7'39″ = 2.33765138048 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 50 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 50**2 + c**2 -2 * 90 * c * cos (30° ) ; ; ; ; c**2 -86.603c -5600 =0 ; ; p=1; q=-86.6025403784; r=-5600 ; ; D = q**2 - 4pr = 86.603**2 - 4 * 1 * (-5600) = 29900 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 86.6 ± sqrt{ 29900 } }{ 2 } = fraction{ 86.6 ± 10 sqrt{ 299 } }{ 2 } ; ; c_{1,2} = 43.3012701892 ± 86.458082329 ; ;
c_{1} = 129.759352518 ; ; c_{2} = -43.1568121397 ; ; ; ; (c -129.759352518) (c +43.1568121397) = 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 = 50 ; ; b = 90 ; ; c = 129.76 ; ;

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

p = a+b+c = 50+90+129.76 = 269.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 269.76 }{ 2 } = 134.88 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.88 * (134.88-50)(134.88-90)(134.88-129.76) } ; ; T = sqrt{ 2630857.74 } = 1621.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1621.99 }{ 50 } = 64.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1621.99 }{ 90 } = 36.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1621.99 }{ 129.76 } = 25 ; ;

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{ 50**2-90**2-129.76**2 }{ 2 * 90 * 129.76 } ) = 16° 7'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-50**2-129.76**2 }{ 2 * 50 * 129.76 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 129.76**2-50**2-90**2 }{ 2 * 90 * 50 } ) = 133° 52'21" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1621.99 }{ 134.88 } = 12.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 16° 7'39" } = 90 ; ;




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