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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 60**2 = 75**2 + c**2 -2 * 60 * c * cos (20° ) ; ; ; ; c**2 -140.954c +2025 =0 ; ; p=1; q=-140.953893118; 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} = 70.4769465589 ± 54.2402064549 ; ; c_{1} = 124.717153014 ; ;
c_{2} = 16.236740104 ; ; ; ; (c -124.717153014) (c -16.236740104) = 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 = 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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