Triangle calculator SSA

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Triangle has two solutions with side c=163.2888063328 and with side c=11.63658780995

#1 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 163.2888063328

Area: T = 3958.181119865
Perimeter: p = 353.2888063328
Semiperimeter: s = 176.6444031664

Angle ∠ A = α = 32.59436460413° = 32°35'37″ = 0.56988664387 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 118.4066353959° = 118°24'23″ = 2.06765807319 rad

Height: ha = 79.1643623973
Height: hb = 87.96595821922
Height: hc = 48.48109620246

Median: ma = 121.9989736506
Median: mb = 127.697689038
Median: mc = 48.8298803934

Inradius: r = 22.40876701678
Circumradius: R = 92.82199402832

Vertex coordinates: A[163.2888063328; 0] B[0; 0] C[87.46219707139; 48.48109620246]
Centroid: CG[83.58333446808; 16.16603206749]
Coordinates of the circumscribed circle: U[81.64440316642; -44.15664650737]
Coordinates of the inscribed circle: I[86.64440316642; 22.40876701678]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.4066353959° = 147°24'23″ = 0.56988664387 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 61.59436460413° = 61°35'37″ = 2.06765807319 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 = 100 ; ; b = 90 ; ; c = 163.29 ; ;

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

p = a+b+c = 100+90+163.29 = 353.29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 353.29 }{ 2 } = 176.64 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 176.64 * (176.64-100)(176.64-90)(176.64-163.29) } ; ; T = sqrt{ 15667198.4 } = 3958.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3958.18 }{ 100 } = 79.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3958.18 }{ 90 } = 87.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3958.18 }{ 163.29 } = 48.48 ; ;

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{ 100**2-90**2-163.29**2 }{ 2 * 90 * 163.29 } ) = 32° 35'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-163.29**2 }{ 2 * 100 * 163.29 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 163.29**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 118° 24'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3958.18 }{ 176.64 } = 22.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 32° 35'37" } = 92.82 ; ;





#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 11.63658780995

Area: T = 282.0599282134
Perimeter: p = 201.63658781
Semiperimeter: s = 100.818793905

Angle ∠ A = α = 147.4066353959° = 147°24'23″ = 2.57327262149 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 3.59436460413° = 3°35'37″ = 0.06327209556 rad

Height: ha = 5.64111856427
Height: hb = 6.26879840474
Height: hc = 48.48109620246

Median: ma = 40.22106020538
Median: mb = 55.16106456595
Median: mc = 94.95334179754

Inradius: r = 2.79877092648
Circumradius: R = 92.82199402832

Vertex coordinates: A[11.63658780995; 0] B[0; 0] C[87.46219707139; 48.48109620246]
Centroid: CG[33.03326162712; 16.16603206749]
Coordinates of the circumscribed circle: U[5.81879390498; 92.63774270983]
Coordinates of the inscribed circle: I[10.81879390498; 2.79877092648]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 32.59436460413° = 32°35'37″ = 2.57327262149 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 176.4066353959° = 176°24'23″ = 0.06327209556 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 100**2 + c**2 -2 * 90 * c * cos (29° ) ; ; ; ; c**2 -174.924c +1900 =0 ; ; p=1; q=-174.923941428; r=1900 ; ; D = q**2 - 4pr = 174.924**2 - 4 * 1 * 1900 = 22998.3852847 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 174.92 ± sqrt{ 22998.39 } }{ 2 } ; ; c_{1,2} = 87.4619707139 ± 75.8260926144 ; ; c_{1} = 163.288063328 ; ;
c_{2} = 11.6358780995 ; ; ; ; (c -163.288063328) (c -11.6358780995) = 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 = 100 ; ; b = 90 ; ; c = 11.64 ; ;

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

p = a+b+c = 100+90+11.64 = 201.64 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.64 }{ 2 } = 100.82 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.82 * (100.82-100)(100.82-90)(100.82-11.64) } ; ; T = sqrt{ 79557.44 } = 282.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 282.06 }{ 100 } = 5.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 282.06 }{ 90 } = 6.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 282.06 }{ 11.64 } = 48.48 ; ;

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{ 100**2-90**2-11.64**2 }{ 2 * 90 * 11.64 } ) = 147° 24'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-11.64**2 }{ 2 * 100 * 11.64 } ) = 29° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.64**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 3° 35'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 282.06 }{ 100.82 } = 2.8 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 147° 24'23" } = 92.82 ; ;




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