Triangle calculator SSA

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Triangle has two solutions with side c=114.8532813742 and with side c=54.85328137424

#1 Acute scalene triangle.

Sides: a = 120   b = 90   c = 114.8532813742

Area: T = 4872.792220614
Perimeter: p = 324.8532813742
Semiperimeter: s = 162.4266406871

Angle ∠ A = α = 70.52987793655° = 70°31'44″ = 1.23109594173 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 64.47112206345° = 64°28'16″ = 1.12552350729 rad

Height: ha = 81.21332034356
Height: hb = 108.2844271248
Height: hc = 84.85328137424

Median: ma = 83.9387979558
Median: mb = 108.4922324209
Median: mc = 89.17551523344

Inradius: r = 30
Circumradius: R = 63.64396103068

Vertex coordinates: A[114.8532813742; 0] B[0; 0] C[84.85328137424; 84.85328137424]
Centroid: CG[66.56985424949; 28.28442712475]
Coordinates of the circumscribed circle: U[57.42664068712; 27.42664068712]
Coordinates of the inscribed circle: I[72.42664068712; 30]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 109.4711220634° = 109°28'16″ = 1.23109594173 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 115.5298779366° = 115°31'44″ = 1.12552350729 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 = 120 ; ; b = 90 ; ; c = 114.85 ; ;

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

p = a+b+c = 120+90+114.85 = 324.85 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 324.85 }{ 2 } = 162.43 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 162.43 * (162.43-120)(162.43-90)(162.43-114.85) } ; ; T = sqrt{ 23744103.88 } = 4872.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4872.79 }{ 120 } = 81.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4872.79 }{ 90 } = 108.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4872.79 }{ 114.85 } = 84.85 ; ;

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{ 120**2-90**2-114.85**2 }{ 2 * 90 * 114.85 } ) = 70° 31'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-114.85**2 }{ 2 * 120 * 114.85 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 114.85**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 64° 28'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4872.79 }{ 162.43 } = 30 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 70° 31'44" } = 63.64 ; ;





#2 Obtuse scalene triangle.

Sides: a = 120   b = 90   c = 54.85328137424

Area: T = 2327.208779386
Perimeter: p = 264.8532813742
Semiperimeter: s = 132.4266406871

Angle ∠ A = α = 109.4711220634° = 109°28'16″ = 1.91106332362 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 25.52987793655° = 25°31'44″ = 0.44655612539 rad

Height: ha = 38.78767965644
Height: hb = 51.71657287525
Height: hc = 84.85328137424

Median: ma = 44.20987727462
Median: mb = 81.72876916824
Median: mc = 102.4598734162

Inradius: r = 17.57435931288
Circumradius: R = 63.64396103068

Vertex coordinates: A[54.85328137424; 0] B[0; 0] C[84.85328137424; 84.85328137424]
Centroid: CG[46.56985424949; 28.28442712475]
Coordinates of the circumscribed circle: U[27.42664068712; 57.42664068712]
Coordinates of the inscribed circle: I[42.42664068712; 17.57435931288]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 70.52987793655° = 70°31'44″ = 1.91106332362 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 154.4711220634° = 154°28'16″ = 0.44655612539 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 120**2 + c**2 -2 * 90 * c * cos (45° ) ; ; ; ; c**2 -169.706c +6300 =0 ; ; p=1; q=-169.705627485; r=6300 ; ; D = q**2 - 4pr = 169.706**2 - 4 * 1 * 6300 = 3600 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 169.71 ± sqrt{ 3600 } }{ 2 } ; ; c_{1,2} = 84.8528137424 ± 30 ; ; c_{1} = 114.852813742 ; ;
c_{2} = 54.8528137424 ; ; ; ; (c -114.852813742) (c -54.8528137424) = 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 = 120 ; ; b = 90 ; ; c = 54.85 ; ;

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

p = a+b+c = 120+90+54.85 = 264.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 264.85 }{ 2 } = 132.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.43 * (132.43-120)(132.43-90)(132.43-54.85) } ; ; T = sqrt{ 5415896.12 } = 2327.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2327.21 }{ 120 } = 38.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2327.21 }{ 90 } = 51.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2327.21 }{ 54.85 } = 84.85 ; ;

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{ 120**2-90**2-54.85**2 }{ 2 * 90 * 54.85 } ) = 109° 28'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-120**2-54.85**2 }{ 2 * 120 * 54.85 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54.85**2-120**2-90**2 }{ 2 * 90 * 120 } ) = 25° 31'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2327.21 }{ 132.43 } = 17.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 109° 28'16" } = 63.64 ; ;




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