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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 67**2 = 93**2 + c**2 -2 * 67 * c * cos (31° ) ; ; ; ; c**2 -159.433c +4160 =0 ; ; p=1; q=-159.433117931; 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} = 79.7165589653 ± 46.847943106 ; ; c_{1} = 126.564502071 ; ;
c_{2} = 32.8686158592 ; ; ; ; (c -126.564502071) (c -32.8686158592) = 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 = 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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