Triangle calculator SSA

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Triangle has two solutions with side c=38.96876297655 and with side c=6.62197502017

#1 Obtuse scalene triangle.

Sides: a = 23.5   b = 17.155   c = 38.96876297655

Area: T = 111.4122155518
Perimeter: p = 79.62326297655
Semiperimeter: s = 39.81113148828

Angle ∠ A = α = 19.4710712896° = 19°28'15″ = 0.34398280477 rad
Angle ∠ B = β = 14.083° = 14°4'59″ = 0.24657947186 rad
Angle ∠ C = γ = 146.4466287104° = 146°26'47″ = 2.55659698873 rad

Height: ha = 9.4821885576
Height: hb = 12.98988843507
Height: hc = 5.71881900048

Median: ma = 27.71986326732
Median: mb = 31.01327325226
Median: mc = 6.60770394364

Inradius: r = 2.79985047931
Circumradius: R = 35.25108835541

Vertex coordinates: A[38.96876297655; 0] B[0; 0] C[22.79436899836; 5.71881900048]
Centroid: CG[20.5877106583; 1.90660633349]
Coordinates of the circumscribed circle: U[19.48438148828; -29.37769594915]
Coordinates of the inscribed circle: I[22.65663148828; 2.79985047931]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5299287104° = 160°31'45″ = 0.34398280477 rad
∠ B' = β' = 165.917° = 165°55'1″ = 0.24657947186 rad
∠ C' = γ' = 33.5543712896° = 33°33'13″ = 2.55659698873 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 = 23.5 ; ; b = 17.16 ; ; c = 38.97 ; ;

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

p = a+b+c = 23.5+17.16+38.97 = 79.62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79.62 }{ 2 } = 39.81 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.81 * (39.81-23.5)(39.81-17.16)(39.81-38.97) } ; ; T = sqrt{ 12412.67 } = 111.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.41 }{ 23.5 } = 9.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.41 }{ 17.16 } = 12.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.41 }{ 38.97 } = 5.72 ; ;

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{ 23.5**2-17.16**2-38.97**2 }{ 2 * 17.16 * 38.97 } ) = 19° 28'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17.16**2-23.5**2-38.97**2 }{ 2 * 23.5 * 38.97 } ) = 14° 4'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.97**2-23.5**2-17.16**2 }{ 2 * 17.16 * 23.5 } ) = 146° 26'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.41 }{ 39.81 } = 2.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23.5 }{ 2 * sin 19° 28'15" } = 35.25 ; ;





#2 Obtuse scalene triangle.

Sides: a = 23.5   b = 17.155   c = 6.62197502017

Area: T = 18.92664947188
Perimeter: p = 47.27547502017
Semiperimeter: s = 23.63773751008

Angle ∠ A = α = 160.5299287104° = 160°31'45″ = 2.80217646058 rad
Angle ∠ B = β = 14.083° = 14°4'59″ = 0.24657947186 rad
Angle ∠ C = γ = 5.3887712896° = 5°23'16″ = 0.09440333292 rad

Height: ha = 1.6110765508
Height: hb = 2.20765280931
Height: hc = 5.71881900048

Median: ma = 5.56773206183
Median: mb = 14.98220572725
Median: mc = 20.30655839442

Inradius: r = 0.80107020508
Circumradius: R = 35.25108835541

Vertex coordinates: A[6.62197502017; 0] B[0; 0] C[22.79436899836; 5.71881900048]
Centroid: CG[9.80444800618; 1.90660633349]
Coordinates of the circumscribed circle: U[3.31098751008; 35.09551494963]
Coordinates of the inscribed circle: I[6.48223751008; 0.80107020508]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.4710712896° = 19°28'15″ = 2.80217646058 rad
∠ B' = β' = 165.917° = 165°55'1″ = 0.24657947186 rad
∠ C' = γ' = 174.6122287104° = 174°36'44″ = 0.09440333292 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 23.5 ; ; b = 17.16 ; ; beta = 14° 4'59" ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 17.16**2 = 23.5**2 + c**2 -2 * 17.16 * c * cos (14° 4'59") ; ; ; ; c**2 -45.587c +257.956 =0 ; ; p=1; q=-45.5873799672; r=257.955975 ; ; D = q**2 - 4pr = 45.587**2 - 4 * 1 * 257.956 = 1046.38531228 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 45.59 ± sqrt{ 1046.39 } }{ 2 } ; ; c_{1,2} = 22.7936899836 ± 16.1739397819 ; ;
c_{1} = 38.9676297655 ; ; c_{2} = 6.6197502017 ; ; ; ; (c -38.9676297655) (c -6.6197502017) = 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 = 23.5 ; ; b = 17.16 ; ; c = 6.62 ; ;

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

p = a+b+c = 23.5+17.16+6.62 = 47.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.27 }{ 2 } = 23.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.64 * (23.64-23.5)(23.64-17.16)(23.64-6.62) } ; ; T = sqrt{ 358.21 } = 18.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.93 }{ 23.5 } = 1.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.93 }{ 17.16 } = 2.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.93 }{ 6.62 } = 5.72 ; ;

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{ 23.5**2-17.16**2-6.62**2 }{ 2 * 17.16 * 6.62 } ) = 160° 31'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17.16**2-23.5**2-6.62**2 }{ 2 * 23.5 * 6.62 } ) = 14° 4'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.62**2-23.5**2-17.16**2 }{ 2 * 17.16 * 23.5 } ) = 5° 23'16" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.93 }{ 23.64 } = 0.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23.5 }{ 2 * sin 160° 31'45" } = 35.25 ; ;




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