Triangle calculator SSA

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Triangle has two solutions with side c=32.27552863135 and with side c=11.70879550071

#1 Obtuse scalene triangle.

Sides: a = 22.51   b = 11.35   c = 32.27552863135

Area: T = 77.50989517975
Perimeter: p = 66.13552863135
Semiperimeter: s = 33.06876431568

Angle ∠ A = α = 25.03549482603° = 25°2'6″ = 0.43769422752 rad
Angle ∠ B = β = 12.32° = 12°19'12″ = 0.21550245638 rad
Angle ∠ C = γ = 142.645505174° = 142°38'42″ = 2.49896258145 rad

Height: ha = 6.88766238825
Height: hb = 13.65879650745
Height: hc = 4.80329908113

Median: ma = 21.41545576258
Median: mb = 27.24395205228
Median: mc = 7.57221709797

Inradius: r = 2.34439515006
Circumradius: R = 26.59768133231

Vertex coordinates: A[32.27552863135; 0] B[0; 0] C[21.99216206603; 4.80329908113]
Centroid: CG[18.08989689913; 1.60109969371]
Coordinates of the circumscribed circle: U[16.13876431568; -21.14215929459]
Coordinates of the inscribed circle: I[21.71876431568; 2.34439515006]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.965505174° = 154°57'54″ = 0.43769422752 rad
∠ B' = β' = 167.68° = 167°40'48″ = 0.21550245638 rad
∠ C' = γ' = 37.35549482603° = 37°21'18″ = 2.49896258145 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 22.51 ; ; b = 11.35 ; ; beta = 12° 19'12" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.35**2 = 22.51**2 + c**2 -2 * 22.51 * c * cos (12° 19'12") ; ; ; ; c**2 -43.983c +377.878 =0 ; ; p=1; q=-43.983; r=377.878 ; ; D = q**2 - 4pr = 43.983**2 - 4 * 1 * 377.878 = 423.015117067 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 43.98 ± sqrt{ 423.02 } }{ 2 } ; ; c_{1,2} = 21.99162066 ± 10.2836656532 ; ;
c_{1} = 32.2752863132 ; ; c_{2} = 11.7079550068 ; ; ; ; text{ Factored form: } ; ; (c -32.2752863132) (c -11.7079550068) = 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 = 22.51 ; ; b = 11.35 ; ; c = 32.28 ; ;

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

p = a+b+c = 22.51+11.35+32.28 = 66.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66.14 }{ 2 } = 33.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.07 * (33.07-22.51)(33.07-11.35)(33.07-32.28) } ; ; T = sqrt{ 6007.64 } = 77.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.51 }{ 22.51 } = 6.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.51 }{ 11.35 } = 13.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.51 }{ 32.28 } = 4.8 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 11.35**2+32.28**2-22.51**2 }{ 2 * 11.35 * 32.28 } ) = 25° 2'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.51**2+32.28**2-11.35**2 }{ 2 * 22.51 * 32.28 } ) = 12° 19'12" ; ; gamma = 180° - alpha - beta = 180° - 25° 2'6" - 12° 19'12" = 142° 38'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.51 }{ 33.07 } = 2.34 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.51 }{ 2 * sin 25° 2'6" } = 26.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.35**2+2 * 32.28**2 - 22.51**2 } }{ 2 } = 21.415 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.28**2+2 * 22.51**2 - 11.35**2 } }{ 2 } = 27.24 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.35**2+2 * 22.51**2 - 32.28**2 } }{ 2 } = 7.572 ; ;







#2 Obtuse scalene triangle.

Sides: a = 22.51   b = 11.35   c = 11.70879550071

Area: T = 28.1176600159
Perimeter: p = 45.56879550071
Semiperimeter: s = 22.78439775036

Angle ∠ A = α = 154.965505174° = 154°57'54″ = 2.70546503784 rad
Angle ∠ B = β = 12.32° = 12°19'12″ = 0.21550245638 rad
Angle ∠ C = γ = 12.71549482603° = 12°42'54″ = 0.22219177114 rad

Height: ha = 2.49881430617
Height: hb = 4.95444669884
Height: hc = 4.80329908113

Median: ma = 2.50548613184
Median: mb = 17.02200625799
Median: mc = 16.83772280197

Inradius: r = 1.23440514361
Circumradius: R = 26.59768133231

Vertex coordinates: A[11.70879550071; 0] B[0; 0] C[21.99216206603; 4.80329908113]
Centroid: CG[11.23331918891; 1.60109969371]
Coordinates of the circumscribed circle: U[5.85439775036; 25.94545837571]
Coordinates of the inscribed circle: I[11.43439775036; 1.23440514361]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 25.03549482603° = 25°2'6″ = 2.70546503784 rad
∠ B' = β' = 167.68° = 167°40'48″ = 0.21550245638 rad
∠ C' = γ' = 167.285505174° = 167°17'6″ = 0.22219177114 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 22.51 ; ; b = 11.35 ; ; beta = 12° 19'12" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 11.35**2 = 22.51**2 + c**2 -2 * 22.51 * c * cos (12° 19'12") ; ; ; ; c**2 -43.983c +377.878 =0 ; ; p=1; q=-43.983; r=377.878 ; ; D = q**2 - 4pr = 43.983**2 - 4 * 1 * 377.878 = 423.015117067 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 43.98 ± sqrt{ 423.02 } }{ 2 } ; ; c_{1,2} = 21.99162066 ± 10.2836656532 ; ; : Nr. 1
c_{1} = 32.2752863132 ; ; c_{2} = 11.7079550068 ; ; ; ; text{ Factored form: } ; ; (c -32.2752863132) (c -11.7079550068) = 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 = 22.51 ; ; b = 11.35 ; ; c = 11.71 ; ;

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

p = a+b+c = 22.51+11.35+11.71 = 45.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.57 }{ 2 } = 22.78 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.78 * (22.78-22.51)(22.78-11.35)(22.78-11.71) } ; ; T = sqrt{ 790.54 } = 28.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.12 }{ 22.51 } = 2.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.12 }{ 11.35 } = 4.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.12 }{ 11.71 } = 4.8 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 11.35**2+11.71**2-22.51**2 }{ 2 * 11.35 * 11.71 } ) = 154° 57'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.51**2+11.71**2-11.35**2 }{ 2 * 22.51 * 11.71 } ) = 12° 19'12" ; ; gamma = 180° - alpha - beta = 180° - 154° 57'54" - 12° 19'12" = 12° 42'54" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.12 }{ 22.78 } = 1.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.51 }{ 2 * sin 154° 57'54" } = 26.6 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.35**2+2 * 11.71**2 - 22.51**2 } }{ 2 } = 2.505 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.71**2+2 * 22.51**2 - 11.35**2 } }{ 2 } = 17.02 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.35**2+2 * 22.51**2 - 11.71**2 } }{ 2 } = 16.837 ; ;
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