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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 11.35**2 = 22.51**2 + c**2 -2 * 11.35 * c * cos (12° 19'12") ; ; ; ; c**2 -43.983c +377.878 =0 ; ; p=1; q=-43.9832413206; r=377.8776 ; ; 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} = 21.9916206603 ± 10.2836656532 ; ;
c_{1} = 32.2752863135 ; ; c_{2} = 11.7079550071 ; ; ; ; (c -32.2752863135) (c -11.7079550071) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 22.51**2-11.35**2-11.71**2 }{ 2 * 11.35 * 11.71 } ) = 154° 57'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.35**2-22.51**2-11.71**2 }{ 2 * 22.51 * 11.71 } ) = 12° 19'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.71**2-22.51**2-11.35**2 }{ 2 * 11.35 * 22.51 } ) = 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 ; ;




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