Triangle calculator SSA

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Triangle has two solutions with side c=126.5654502071 and with side c=32.86986158592

#1 Obtuse scalene triangle.

Sides: a = 93   b = 67   c = 126.5654502071

Area: T = 3031.127749369
Perimeter: p = 286.5654502071
Semiperimeter: s = 143.2822251036

Angle ∠ A = α = 45.63552991043° = 45°38'7″ = 0.79664862245 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 103.3654700896° = 103°21'53″ = 1.8044054361 rad

Height: ha = 65.18655374988
Height: hb = 90.48114177222
Height: hc = 47.89985409666

Median: ma = 89.95329687797
Median: mb = 105.8854543689
Median: mc = 50.6399477721

Inradius: r = 21.15549404883
Circumradius: R = 65.04437348847

Vertex coordinates: A[126.5654502071; 0] B[0; 0] C[79.71765589653; 47.89985409666]
Centroid: CG[68.76603536789; 15.96661803222]
Coordinates of the circumscribed circle: U[63.28222510357; -15.03547647675]
Coordinates of the inscribed circle: I[76.28222510357; 21.15549404883]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.3654700896° = 134°21'53″ = 0.79664862245 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 76.63552991043° = 76°38'7″ = 1.8044054361 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 93 ; ; b = 67 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 67**2 = 93**2 + c**2 -2 * 93 * c * cos (31° ) ; ; ; ; c**2 -159.433c +4160 =0 ; ; p=1; q=-159.433; r=4160 ; ; D = q**2 - 4pr = 159.433**2 - 4 * 1 * 4160 = 8778.91909307 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 159.43 ± sqrt{ 8778.92 } }{ 2 } ; ; c_{1,2} = 79.71655897 ± 46.847943106 ; ; c_{1} = 126.564502076 ; ;
c_{2} = 32.868615864 ; ; ; ; (c -126.564502076) (c -32.868615864) = 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 = 93 ; ; b = 67 ; ; c = 126.56 ; ;

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

p = a+b+c = 93+67+126.56 = 286.56 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 286.56 }{ 2 } = 143.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 143.28 * (143.28-93)(143.28-67)(143.28-126.56) } ; ; T = sqrt{ 9187733.88 } = 3031.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3031.13 }{ 93 } = 65.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3031.13 }{ 67 } = 90.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3031.13 }{ 126.56 } = 47.9 ; ;

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{ 93**2-67**2-126.56**2 }{ 2 * 67 * 126.56 } ) = 45° 38'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-93**2-126.56**2 }{ 2 * 93 * 126.56 } ) = 31° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 126.56**2-93**2-67**2 }{ 2 * 67 * 93 } ) = 103° 21'53" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3031.13 }{ 143.28 } = 21.15 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 93 }{ 2 * sin 45° 38'7" } = 65.04 ; ;





#2 Obtuse scalene triangle.

Sides: a = 93   b = 67   c = 32.86986158592

Area: T = 787.1799371625
Perimeter: p = 192.8698615859
Semiperimeter: s = 96.43443079296

Angle ∠ A = α = 134.3654700896° = 134°21'53″ = 2.34551064291 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 14.63552991043° = 14°38'7″ = 0.25554341564 rad

Height: ha = 16.92985886371
Height: hb = 23.49878916903
Height: hc = 47.89985409666

Median: ma = 24.94884058459
Median: mb = 61.17553459676
Median: mc = 79.36656948743

Inradius: r = 8.16328560263
Circumradius: R = 65.04437348847

Vertex coordinates: A[32.86986158592; 0] B[0; 0] C[79.71765589653; 47.89985409666]
Centroid: CG[37.52883916082; 15.96661803222]
Coordinates of the circumscribed circle: U[16.43443079296; 62.93333057342]
Coordinates of the inscribed circle: I[29.43443079296; 8.16328560263]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 45.63552991043° = 45°38'7″ = 2.34551064291 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 165.3654700896° = 165°21'53″ = 0.25554341564 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 93 ; ; b = 67 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 67**2 = 93**2 + c**2 -2 * 93 * c * cos (31° ) ; ; ; ; c**2 -159.433c +4160 =0 ; ; p=1; q=-159.433; r=4160 ; ; D = q**2 - 4pr = 159.433**2 - 4 * 1 * 4160 = 8778.91909307 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 159.43 ± sqrt{ 8778.92 } }{ 2 } ; ; c_{1,2} = 79.71655897 ± 46.847943106 ; ; c_{1} = 126.564502076 ; ; : Nr. 1
c_{2} = 32.868615864 ; ; ; ; (c -126.564502076) (c -32.868615864) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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 = 93 ; ; b = 67 ; ; c = 32.87 ; ;

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

p = a+b+c = 93+67+32.87 = 192.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 192.87 }{ 2 } = 96.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 96.43 * (96.43-93)(96.43-67)(96.43-32.87) } ; ; T = sqrt{ 619651.36 } = 787.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 787.18 }{ 93 } = 16.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 787.18 }{ 67 } = 23.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 787.18 }{ 32.87 } = 47.9 ; ;

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{ 93**2-67**2-32.87**2 }{ 2 * 67 * 32.87 } ) = 134° 21'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-93**2-32.87**2 }{ 2 * 93 * 32.87 } ) = 31° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 32.87**2-93**2-67**2 }{ 2 * 67 * 93 } ) = 14° 38'7" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 787.18 }{ 96.43 } = 8.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 93 }{ 2 * sin 134° 21'53" } = 65.04 ; ;




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