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?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (29° ) ; ; ; ; c**2 -174.924c +1900 =0 ; ; p=1; q=-174.924; 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} = 87.46197071 ± 75.8260926144 ; ; c_{1} = 163.288063324 ; ; c_{2} = 11.6358780956 ; ; ; ; text{ Factored form: } ; ; (c -163.288063324) (c -11.6358780956) = 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 = 163.29 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 90**2+163.29**2-100**2 }{ 2 * 90 * 163.29 } ) = 32° 35'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+163.29**2-90**2 }{ 2 * 100 * 163.29 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 32° 35'37" - 29° = 118° 24'23" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 163.29**2 - 100**2 } }{ 2 } = 121.99 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 163.29**2+2 * 100**2 - 90**2 } }{ 2 } = 127.697 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 163.29**2 } }{ 2 } = 48.829 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 29° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (29° ) ; ; ; ; c**2 -174.924c +1900 =0 ; ; p=1; q=-174.924; 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} = 87.46197071 ± 75.8260926144 ; ; c_{1} = 163.288063324 ; ; c_{2} = 11.6358780956 ; ; ; ; text{ Factored form: } ; ; (c -163.288063324) (c -11.6358780956) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+11.64**2-100**2 }{ 2 * 90 * 11.64 } ) = 147° 24'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+11.64**2-90**2 }{ 2 * 100 * 11.64 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 147° 24'23" - 29° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 11.64**2 - 100**2 } }{ 2 } = 40.221 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.64**2+2 * 100**2 - 90**2 } }{ 2 } = 55.161 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 11.64**2 } }{ 2 } = 94.953 ; ;
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