Triangle calculator SSA

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Triangle has two solutions with side c=90.45987638629 and with side c=87.84439762016

#1 Obtuse scalene triangle.

Sides: a = 89.6   b = 9.05   c = 90.45987638629

Area: T = 405.0322136466
Perimeter: p = 189.1098763863
Semiperimeter: s = 94.55443819315

Angle ∠ A = α = 81.69437924986° = 81°41'38″ = 1.42658256575 rad
Angle ∠ B = β = 5.736° = 5°44'10″ = 0.11001120859 rad
Angle ∠ C = γ = 92.57702075014° = 92°34'13″ = 1.61656549102 rad

Height: ha = 9.04108959033
Height: hb = 89.51098644124
Height: hc = 8.95550667988

Median: ma = 46.10110328496
Median: mb = 89.91766189022
Median: mc = 44.82655982682

Inradius: r = 4.28435892763
Circumradius: R = 45.27549274916

Vertex coordinates: A[90.45987638629; 0] B[0; 0] C[89.15113700323; 8.95550667988]
Centroid: CG[59.87700446317; 2.98550222663]
Coordinates of the circumscribed circle: U[45.22993819315; -2.03302880269]
Coordinates of the inscribed circle: I[85.50443819315; 4.28435892763]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.30662075014° = 98°18'22″ = 1.42658256575 rad
∠ B' = β' = 174.264° = 174°15'50″ = 0.11001120859 rad
∠ C' = γ' = 87.43297924986° = 87°25'47″ = 1.61656549102 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 89.6 ; ; b = 9.05 ; ; beta = 5° 44'10" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.05**2 = 89.6**2 + c**2 -2 * 89.6 * c * cos (5° 44'10") ; ; ; ; c**2 -178.303c +7946.258 =0 ; ; p=1; q=-178.303; r=7946.258 ; ; D = q**2 - 4pr = 178.303**2 - 4 * 1 * 7946.258 = 6.83711451371 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 178.3 ± sqrt{ 6.84 } }{ 2 } ; ; c_{1,2} = 89.15137003 ± 1.30739383065 ; ;
c_{1} = 90.4587638607 ; ; c_{2} = 87.8439761993 ; ; ; ; text{ Factored form: } ; ; (c -90.4587638607) (c -87.8439761993) = 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 = 89.6 ; ; b = 9.05 ; ; c = 90.46 ; ;

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

p = a+b+c = 89.6+9.05+90.46 = 189.11 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 189.11 }{ 2 } = 94.55 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.55 * (94.55-89.6)(94.55-9.05)(94.55-90.46) } ; ; T = sqrt{ 164051.03 } = 405.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 405.03 }{ 89.6 } = 9.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 405.03 }{ 9.05 } = 89.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 405.03 }{ 90.46 } = 8.96 ; ;

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{ 9.05**2+90.46**2-89.6**2 }{ 2 * 9.05 * 90.46 } ) = 81° 41'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89.6**2+90.46**2-9.05**2 }{ 2 * 89.6 * 90.46 } ) = 5° 44'10" ; ; gamma = 180° - alpha - beta = 180° - 81° 41'38" - 5° 44'10" = 92° 34'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 405.03 }{ 94.55 } = 4.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 89.6 }{ 2 * sin 81° 41'38" } = 45.27 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.05**2+2 * 90.46**2 - 89.6**2 } }{ 2 } = 46.101 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90.46**2+2 * 89.6**2 - 9.05**2 } }{ 2 } = 89.917 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.05**2+2 * 89.6**2 - 90.46**2 } }{ 2 } = 44.826 ; ;







#2 Obtuse scalene triangle.

Sides: a = 89.6   b = 9.05   c = 87.84439762016

Area: T = 393.324433738
Perimeter: p = 186.4943976202
Semiperimeter: s = 93.24769881008

Angle ∠ A = α = 98.30662075014° = 98°18'22″ = 1.7165766996 rad
Angle ∠ B = β = 5.736° = 5°44'10″ = 0.11001120859 rad
Angle ∠ C = γ = 75.95877924986° = 75°57'28″ = 1.32657135716 rad

Height: ha = 8.78795611022
Height: hb = 86.92325054984
Height: hc = 8.95550667988

Median: ma = 43.49993485865
Median: mb = 88.61108709609
Median: mc = 46.10773769724

Inradius: r = 4.21880916016
Circumradius: R = 45.27549274916

Vertex coordinates: A[87.84439762016; 0] B[0; 0] C[89.15113700323; 8.95550667988]
Centroid: CG[58.99884487446; 2.98550222663]
Coordinates of the circumscribed circle: U[43.92219881008; 10.98553548257]
Coordinates of the inscribed circle: I[84.19769881008; 4.21880916016]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 81.69437924986° = 81°41'38″ = 1.7165766996 rad
∠ B' = β' = 174.264° = 174°15'50″ = 0.11001120859 rad
∠ C' = γ' = 104.0422207501° = 104°2'32″ = 1.32657135716 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 89.6 ; ; b = 9.05 ; ; beta = 5° 44'10" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.05**2 = 89.6**2 + c**2 -2 * 89.6 * c * cos (5° 44'10") ; ; ; ; c**2 -178.303c +7946.258 =0 ; ; p=1; q=-178.303; r=7946.258 ; ; D = q**2 - 4pr = 178.303**2 - 4 * 1 * 7946.258 = 6.83711451371 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 178.3 ± sqrt{ 6.84 } }{ 2 } ; ; c_{1,2} = 89.15137003 ± 1.30739383065 ; ; : Nr. 1
c_{1} = 90.4587638607 ; ; c_{2} = 87.8439761993 ; ; ; ; text{ Factored form: } ; ; (c -90.4587638607) (c -87.8439761993) = 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 = 89.6 ; ; b = 9.05 ; ; c = 87.84 ; ;

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

p = a+b+c = 89.6+9.05+87.84 = 186.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 186.49 }{ 2 } = 93.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 93.25 * (93.25-89.6)(93.25-9.05)(93.25-87.84) } ; ; T = sqrt{ 154704.03 } = 393.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 393.32 }{ 89.6 } = 8.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 393.32 }{ 9.05 } = 86.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 393.32 }{ 87.84 } = 8.96 ; ;

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{ 9.05**2+87.84**2-89.6**2 }{ 2 * 9.05 * 87.84 } ) = 98° 18'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89.6**2+87.84**2-9.05**2 }{ 2 * 89.6 * 87.84 } ) = 5° 44'10" ; ; gamma = 180° - alpha - beta = 180° - 98° 18'22" - 5° 44'10" = 75° 57'28" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 393.32 }{ 93.25 } = 4.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 89.6 }{ 2 * sin 98° 18'22" } = 45.27 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.05**2+2 * 87.84**2 - 89.6**2 } }{ 2 } = 43.499 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.84**2+2 * 89.6**2 - 9.05**2 } }{ 2 } = 88.611 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.05**2+2 * 89.6**2 - 87.84**2 } }{ 2 } = 46.107 ; ;
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