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?

1. Use Law of Cosines

a = 120 ; ; b = 90 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (45° ) ; ; ; ; c**2 -169.706c +6300 =0 ; ; p=1; q=-169.706; 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} = 84.85281374 ± 30 ; ; c_{1} = 114.85281374 ; ; c_{2} = 54.85281374 ; ;
 ; ; (c -114.85281374) (c -54.85281374) = 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 = 114.85 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-114.85**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 90**2 = 120**2 + c**2 -2 * 120 * c * cos (45° ) ; ; ; ; c**2 -169.706c +6300 =0 ; ; p=1; q=-169.706; 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} = 84.85281374 ± 30 ; ; c_{1} = 114.85281374 ; ; c_{2} = 54.85281374 ; ; : Nr. 1
 ; ; (c -114.85281374) (c -54.85281374) = 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 = 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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