Triangle calculator SSA

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

Sides: a = 38   b = 49   c = 76.89768917557

Area: T = 774.2343741719
Perimeter: p = 163.8976891756
Semiperimeter: s = 81.94884458778

Angle ∠ A = α = 24.26550163073° = 24°15'54″ = 0.42435044276 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 123.7354983693° = 123°44'6″ = 2.16595828653 rad

Height: ha = 40.7499144301
Height: hb = 31.6011377213
Height: hc = 20.13769320409

Median: ma = 61.61222226579
Median: mb = 55.48325736682
Median: mc = 21.07664562861

Inradius: r = 9.44878148234
Circumradius: R = 46.23334579126

Vertex coordinates: A[76.89768917557; 0] B[0; 0] C[32.22658276539; 20.13769320409]
Centroid: CG[36.37442398032; 6.71223106803]
Coordinates of the circumscribed circle: U[38.44884458778; -25.67658571451]
Coordinates of the inscribed circle: I[32.94884458778; 9.44878148234]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.7354983693° = 155°44'6″ = 0.42435044276 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 56.26550163073° = 56°15'54″ = 2.16595828653 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 38 ; ; b = 49 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 49**2 = 38**2 + c**2 -2 * 49 * c * cos (32° ) ; ; ; ; c**2 -64.452c -957 =0 ; ; p=1; q=-64.4516553079; r=-957 ; ; D = q**2 - 4pr = 64.452**2 - 4 * 1 * (-957) = 7982.01587193 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 64.45 ± sqrt{ 7982.02 } }{ 2 } ; ; c_{1,2} = 32.2258276539 ± 44.6710641017 ; ; c_{1} = 76.8968917557 ; ;
c_{2} = -12.4452364478 ; ; ; ; (c -76.8968917557) (c +12.4452364478) = 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 = 38 ; ; b = 49 ; ; c = 76.9 ; ;

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

p = a+b+c = 38+49+76.9 = 163.9 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 163.9 }{ 2 } = 81.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.95 * (81.95-38)(81.95-49)(81.95-76.9) } ; ; T = sqrt{ 599437.89 } = 774.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 774.23 }{ 38 } = 40.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 774.23 }{ 49 } = 31.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 774.23 }{ 76.9 } = 20.14 ; ;

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{ 38**2-49**2-76.9**2 }{ 2 * 49 * 76.9 } ) = 24° 15'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 49**2-38**2-76.9**2 }{ 2 * 38 * 76.9 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 76.9**2-38**2-49**2 }{ 2 * 49 * 38 } ) = 123° 44'6" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 774.23 }{ 81.95 } = 9.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38 }{ 2 * sin 24° 15'54" } = 46.23 ; ;




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