Triangle calculator SSA

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Triangle has two solutions with side c=124.7177153014 and with side c=16.2376740104

#1 Obtuse scalene triangle.

Sides: a = 75   b = 60   c = 124.7177153014

Area: T = 1599.592169559
Perimeter: p = 259.7177153014
Semiperimeter: s = 129.8598576507

Angle ∠ A = α = 25.31106040172° = 25°18'38″ = 0.44217533758 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 134.6899395983° = 134°41'22″ = 2.35107734274 rad

Height: ha = 42.6565778549
Height: hb = 53.32197231862
Height: hc = 25.65215107494

Median: ma = 90.39332194799
Median: mb = 98.43661931809
Median: mc = 26.90655372745

Inradius: r = 12.31879518721
Circumradius: R = 87.71441320049

Vertex coordinates: A[124.7177153014; 0] B[0; 0] C[70.47769465589; 25.65215107494]
Centroid: CG[65.06546998576; 8.55105035831]
Coordinates of the circumscribed circle: U[62.35985765069; -61.6866115856]
Coordinates of the inscribed circle: I[69.85985765069; 12.31879518721]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.6899395983° = 154°41'22″ = 0.44217533758 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 45.31106040172° = 45°18'38″ = 2.35107734274 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 75 ; ; b = 60 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 75**2 + c**2 -2 * 75 * c * cos (20° ) ; ; ; ; c**2 -140.954c +2025 =0 ; ; p=1; q=-140.954; r=2025 ; ; D = q**2 - 4pr = 140.954**2 - 4 * 1 * 2025 = 11767.9999851 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 140.95 ± sqrt{ 11768 } }{ 2 } ; ; c_{1,2} = 70.47694656 ± 54.2402064549 ; ; c_{1} = 124.717153015 ; ;
c_{2} = 16.2367401051 ; ; ; ; (c -124.717153015) (c -16.2367401051) = 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 = 75 ; ; b = 60 ; ; c = 124.72 ; ;

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

p = a+b+c = 75+60+124.72 = 259.72 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 259.72 }{ 2 } = 129.86 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 129.86 * (129.86-75)(129.86-60)(129.86-124.72) } ; ; T = sqrt{ 2558693.59 } = 1599.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1599.59 }{ 75 } = 42.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1599.59 }{ 60 } = 53.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1599.59 }{ 124.72 } = 25.65 ; ;

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{ 75**2-60**2-124.72**2 }{ 2 * 60 * 124.72 } ) = 25° 18'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-75**2-124.72**2 }{ 2 * 75 * 124.72 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 124.72**2-75**2-60**2 }{ 2 * 60 * 75 } ) = 134° 41'22" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1599.59 }{ 129.86 } = 12.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 75 }{ 2 * sin 25° 18'38" } = 87.71 ; ;





#2 Obtuse scalene triangle.

Sides: a = 75   b = 60   c = 16.2376740104

Area: T = 208.2488456657
Perimeter: p = 151.2376740104
Semiperimeter: s = 75.6188370052

Angle ∠ A = α = 154.6899395983° = 154°41'22″ = 2.76998392778 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 5.31106040172° = 5°18'38″ = 0.09326875254 rad

Height: ha = 5.55332921775
Height: hb = 6.94216152219
Height: hc = 25.65215107494

Median: ma = 22.92552233272
Median: mb = 45.21441113437
Median: mc = 67.42884218093

Inradius: r = 2.75439400349
Circumradius: R = 87.71441320049

Vertex coordinates: A[16.2376740104; 0] B[0; 0] C[70.47769465589; 25.65215107494]
Centroid: CG[28.9054562221; 8.55105035831]
Coordinates of the circumscribed circle: U[8.1188370052; 87.33876266054]
Coordinates of the inscribed circle: I[15.6188370052; 2.75439400349]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 25.31106040172° = 25°18'38″ = 2.76998392778 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 174.6899395983° = 174°41'22″ = 0.09326875254 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 75 ; ; b = 60 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 75**2 + c**2 -2 * 75 * c * cos (20° ) ; ; ; ; c**2 -140.954c +2025 =0 ; ; p=1; q=-140.954; r=2025 ; ; D = q**2 - 4pr = 140.954**2 - 4 * 1 * 2025 = 11767.9999851 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 140.95 ± sqrt{ 11768 } }{ 2 } ; ; c_{1,2} = 70.47694656 ± 54.2402064549 ; ; c_{1} = 124.717153015 ; ; : Nr. 1
c_{2} = 16.2367401051 ; ; ; ; (c -124.717153015) (c -16.2367401051) = 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 = 75 ; ; b = 60 ; ; c = 16.24 ; ;

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

p = a+b+c = 75+60+16.24 = 151.24 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 151.24 }{ 2 } = 75.62 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 75.62 * (75.62-75)(75.62-60)(75.62-16.24) } ; ; T = sqrt{ 43367.42 } = 208.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 208.25 }{ 75 } = 5.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 208.25 }{ 60 } = 6.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 208.25 }{ 16.24 } = 25.65 ; ;

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{ 75**2-60**2-16.24**2 }{ 2 * 60 * 16.24 } ) = 154° 41'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-75**2-16.24**2 }{ 2 * 75 * 16.24 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.24**2-75**2-60**2 }{ 2 * 60 * 75 } ) = 5° 18'38" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 208.25 }{ 75.62 } = 2.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 75 }{ 2 * sin 154° 41'22" } = 87.71 ; ;




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