Triangle calculator SSA

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Triangle has two solutions with side c=806.783292577 and with side c=379.6454871274

#1 Acute scalene triangle.

Sides: a = 890   b = 697   c = 806.783292577

Area: T = 267639.6022111
Perimeter: p = 2393.783292577
Semiperimeter: s = 1196.891146289

Angle ∠ A = α = 72.15768784201° = 72°9'25″ = 1.25993751064 rad
Angle ∠ B = β = 48.2° = 48°12' = 0.84112486995 rad
Angle ∠ C = γ = 59.64331215799° = 59°38'35″ = 1.04109688477 rad

Height: ha = 601.4377308115
Height: hb = 767.9765902759
Height: hc = 663.4743639718

Median: ma = 608.5476501639
Median: mb = 774.6277068115
Median: mc = 690.094407161

Inradius: r = 223.6122257595
Circumradius: R = 467.4876545708

Vertex coordinates: A[806.783292577; 0] B[0; 0] C[593.2143898522; 663.4743639718]
Centroid: CG[466.6665608097; 221.1587879906]
Coordinates of the circumscribed circle: U[403.3911462885; 236.2660445461]
Coordinates of the inscribed circle: I[499.8911462885; 223.6122257595]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.843312158° = 107°50'35″ = 1.25993751064 rad
∠ B' = β' = 131.8° = 131°48' = 0.84112486995 rad
∠ C' = γ' = 120.357687842° = 120°21'25″ = 1.04109688477 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 = 890 ; ; b = 697 ; ; c = 806.78 ; ;

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

p = a+b+c = 890+697+806.78 = 2393.78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2393.78 }{ 2 } = 1196.89 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1196.89 * (1196.89-890)(1196.89-697)(1196.89-806.78) } ; ; T = sqrt{ 71630956618.4 } = 267639.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267639.6 }{ 890 } = 601.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267639.6 }{ 697 } = 767.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267639.6 }{ 806.78 } = 663.47 ; ;

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{ 890**2-697**2-806.78**2 }{ 2 * 697 * 806.78 } ) = 72° 9'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 697**2-890**2-806.78**2 }{ 2 * 890 * 806.78 } ) = 48° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 806.78**2-890**2-697**2 }{ 2 * 697 * 890 } ) = 59° 38'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267639.6 }{ 1196.89 } = 223.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 890 }{ 2 * sin 72° 9'25" } = 467.49 ; ;





#2 Obtuse scalene triangle.

Sides: a = 890   b = 697   c = 379.6454871274

Area: T = 125942.1822272
Perimeter: p = 1966.645487127
Semiperimeter: s = 983.3222435637

Angle ∠ A = α = 107.843312158° = 107°50'35″ = 1.88222175472 rad
Angle ∠ B = β = 48.2° = 48°12' = 0.84112486995 rad
Angle ∠ C = γ = 23.95768784201° = 23°57'25″ = 0.41881264069 rad

Height: ha = 283.0166139937
Height: hb = 361.384359332
Height: hc = 663.4743639718

Median: ma = 341.9721656928
Median: mb = 588.7810828613
Median: mc = 776.4880484577

Inradius: r = 128.0788214945
Circumradius: R = 467.4876545708

Vertex coordinates: A[379.6454871274; 0] B[0; 0] C[593.2143898522; 663.4743639718]
Centroid: CG[324.2866256599; 221.1587879906]
Coordinates of the circumscribed circle: U[189.8222435637; 427.2133194257]
Coordinates of the inscribed circle: I[286.3222435637; 128.0788214945]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.15768784201° = 72°9'25″ = 1.88222175472 rad
∠ B' = β' = 131.8° = 131°48' = 0.84112486995 rad
∠ C' = γ' = 156.043312158° = 156°2'35″ = 0.41881264069 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 890 ; ; b = 697 ; ; beta = 48° 12' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 697**2 = 890**2 + c**2 -2 * 697 * c * cos (48° 12') ; ; ; ; c**2 -1186.428c +306291 =0 ; ; p=1; q=-1186.42779704; r=306291 ; ; D = q**2 - 4pr = 1186.428**2 - 4 * 1 * 306291 = 182446.917599 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1186.43 ± sqrt{ 182446.92 } }{ 2 } ; ; c_{1,2} = 593.213898522 ± 213.569027248 ; ;
c_{1} = 806.78292577 ; ; c_{2} = 379.644871274 ; ; ; ; (c -806.78292577) (c -379.644871274) = 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 = 890 ; ; b = 697 ; ; c = 379.64 ; ;

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

p = a+b+c = 890+697+379.64 = 1966.64 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1966.64 }{ 2 } = 983.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 983.32 * (983.32-890)(983.32-697)(983.32-379.64) } ; ; T = sqrt{ 15861433275.5 } = 125942.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125942.18 }{ 890 } = 283.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125942.18 }{ 697 } = 361.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125942.18 }{ 379.64 } = 663.47 ; ;

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{ 890**2-697**2-379.64**2 }{ 2 * 697 * 379.64 } ) = 107° 50'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 697**2-890**2-379.64**2 }{ 2 * 890 * 379.64 } ) = 48° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 379.64**2-890**2-697**2 }{ 2 * 697 * 890 } ) = 23° 57'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125942.18 }{ 983.32 } = 128.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 890 }{ 2 * sin 107° 50'35" } = 467.49 ; ;




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