Triangle calculator SSA

Please enter two sides and a non-included angle
°


Obtuse scalene triangle.

Sides: a = 40   b = 150   c = 158.8955007674

Area: T = 2986.249932383
Perimeter: p = 348.8955007674
Semiperimeter: s = 174.4487503837

Angle ∠ A = α = 14.51221143405° = 14°30'44″ = 0.25332841767 rad
Angle ∠ B = β = 70° = 1.22217304764 rad
Angle ∠ C = γ = 95.48878856595° = 95°29'16″ = 1.66765780005 rad

Height: ha = 149.3122466191
Height: hb = 39.81766576511
Height: hc = 37.58877048314

Median: ma = 153.2121656645
Median: mb = 88.31108811639
Median: mc = 75.75502088052

Inradius: r = 17.11883264773
Circumradius: R = 79.81333329357

Vertex coordinates: A[158.8955007674; 0] B[0; 0] C[13.6810805733; 37.58877048314]
Centroid: CG[57.52552711358; 12.52992349438]
Coordinates of the circumscribed circle: U[79.44875038373; -7.6332971134]
Coordinates of the inscribed circle: I[24.44875038373; 17.11883264773]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.488788566° = 165°29'16″ = 0.25332841767 rad
∠ B' = β' = 110° = 1.22217304764 rad
∠ C' = γ' = 84.51221143405° = 84°30'44″ = 1.66765780005 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 40 ; ; b = 150 ; ; beta = 70° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 150**2 = 40**2 + c**2 -2 * 150 * c * cos (70° ) ; ; ; ; c**2 -27.362c -20900 =0 ; ; p=1; q=-27.3616114661; r=-20900 ; ; D = q**2 - 4pr = 27.362**2 - 4 * 1 * (-20900) = 84348.657782 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 27.36 ± sqrt{ 84348.66 } }{ 2 } ; ; c_{1,2} = 13.680805733 ± 145.214201941 ; ;
c_{1} = 158.895007675 ; ; c_{2} = -131.533396208 ; ; ; ; (c -158.895007675) (c +131.533396208) = 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 = 40 ; ; b = 150 ; ; c = 158.9 ; ;

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

p = a+b+c = 40+150+158.9 = 348.9 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 348.9 }{ 2 } = 174.45 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 174.45 * (174.45-40)(174.45-150)(174.45-158.9) } ; ; T = sqrt{ 8917685.02 } = 2986.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2986.25 }{ 40 } = 149.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2986.25 }{ 150 } = 39.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2986.25 }{ 158.9 } = 37.59 ; ;

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{ 40**2-150**2-158.9**2 }{ 2 * 150 * 158.9 } ) = 14° 30'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 150**2-40**2-158.9**2 }{ 2 * 40 * 158.9 } ) = 70° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 158.9**2-40**2-150**2 }{ 2 * 150 * 40 } ) = 95° 29'16" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2986.25 }{ 174.45 } = 17.12 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40 }{ 2 * sin 14° 30'44" } = 79.81 ; ;




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