Triangle calculator SSA

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Triangle has two solutions with side c=165.4032888605 and with side c=35.79774401168

#1 Obtuse scalene triangle.

Sides: a = 111   b = 80   c = 165.4032888605

Area: T = 3879.577661044
Perimeter: p = 356.4032888605
Semiperimeter: s = 178.2011444303

Angle ∠ A = α = 35.90107413748° = 35°54'3″ = 0.62765861409 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 119.0999258625° = 119°5'57″ = 2.07986741997 rad

Height: ha = 69.90222812693
Height: hb = 96.98994152611
Height: hc = 46.91106270532

Median: ma = 117.4688326708
Median: mb = 135.0543906939
Median: mc = 50.209927315

Inradius: r = 21.77107360657
Circumradius: R = 94.64880633261

Vertex coordinates: A[165.4032888605; 0] B[0; 0] C[100.6600164361; 46.91106270532]
Centroid: CG[88.66876843221; 15.63768756844]
Coordinates of the circumscribed circle: U[82.70114443026; -46.03296317782]
Coordinates of the inscribed circle: I[98.20114443026; 21.77107360657]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.0999258625° = 144°5'57″ = 0.62765861409 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 60.90107413748° = 60°54'3″ = 2.07986741997 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 111 ; ; b = 80 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 80**2 = 111**2 + c**2 -2 * 111 * c * cos (25° ) ; ; ; ; c**2 -201.2c +5921 =0 ; ; p=1; q=-201.2; r=5921 ; ; D = q**2 - 4pr = 201.2**2 - 4 * 1 * 5921 = 16797.5722779 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 201.2 ± sqrt{ 16797.57 } }{ 2 } ; ; c_{1,2} = 100.60016436 ± 64.8027242442 ; ; c_{1} = 165.402888604 ; ; c_{2} = 35.7974401158 ; ; ; ; text{ Factored form: } ; ; (c -165.402888604) (c -35.7974401158) = 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 = 111 ; ; b = 80 ; ; c = 165.4 ; ;

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

p = a+b+c = 111+80+165.4 = 356.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 356.4 }{ 2 } = 178.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 178.2 * (178.2-111)(178.2-80)(178.2-165.4) } ; ; T = sqrt{ 15051114.68 } = 3879.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3879.58 }{ 111 } = 69.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3879.58 }{ 80 } = 96.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3879.58 }{ 165.4 } = 46.91 ; ;

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{ 80**2+165.4**2-111**2 }{ 2 * 80 * 165.4 } ) = 35° 54'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 111**2+165.4**2-80**2 }{ 2 * 111 * 165.4 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 35° 54'3" - 25° = 119° 5'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3879.58 }{ 178.2 } = 21.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 111 }{ 2 * sin 35° 54'3" } = 94.65 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 165.4**2 - 111**2 } }{ 2 } = 117.468 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 165.4**2+2 * 111**2 - 80**2 } }{ 2 } = 135.054 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 111**2 - 165.4**2 } }{ 2 } = 50.209 ; ;







#2 Obtuse scalene triangle.

Sides: a = 111   b = 80   c = 35.79774401168

Area: T = 839.644018139
Perimeter: p = 226.7977440117
Semiperimeter: s = 113.3998720058

Angle ∠ A = α = 144.0999258625° = 144°5'57″ = 2.51550065127 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 10.90107413748° = 10°54'3″ = 0.19902538279 rad

Height: ha = 15.12986519169
Height: hb = 20.99110045348
Height: hc = 46.91106270532

Median: ma = 27.57767721001
Median: mb = 72.12195421468
Median: mc = 95.08796288396

Inradius: r = 7.40443179761
Circumradius: R = 94.64880633261

Vertex coordinates: A[35.79774401168; 0] B[0; 0] C[100.6600164361; 46.91106270532]
Centroid: CG[45.46658681593; 15.63768756844]
Coordinates of the circumscribed circle: U[17.89987200584; 92.94402588314]
Coordinates of the inscribed circle: I[33.39987200584; 7.40443179761]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 35.90107413748° = 35°54'3″ = 2.51550065127 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 169.0999258625° = 169°5'57″ = 0.19902538279 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 111 ; ; b = 80 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 80**2 = 111**2 + c**2 -2 * 111 * c * cos (25° ) ; ; ; ; c**2 -201.2c +5921 =0 ; ; p=1; q=-201.2; r=5921 ; ; D = q**2 - 4pr = 201.2**2 - 4 * 1 * 5921 = 16797.5722779 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 201.2 ± sqrt{ 16797.57 } }{ 2 } ; ; c_{1,2} = 100.60016436 ± 64.8027242442 ; ; c_{1} = 165.402888604 ; ; c_{2} = 35.7974401158 ; ; ; ; text{ Factored form: } ; ; (c -165.402888604) (c -35.7974401158) = 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 = 111 ; ; b = 80 ; ; c = 35.8 ; ;

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

p = a+b+c = 111+80+35.8 = 226.8 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 226.8 }{ 2 } = 113.4 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 113.4 * (113.4-111)(113.4-80)(113.4-35.8) } ; ; T = sqrt{ 704995.63 } = 839.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 839.64 }{ 111 } = 15.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 839.64 }{ 80 } = 20.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 839.64 }{ 35.8 } = 46.91 ; ;

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{ 80**2+35.8**2-111**2 }{ 2 * 80 * 35.8 } ) = 144° 5'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 111**2+35.8**2-80**2 }{ 2 * 111 * 35.8 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 144° 5'57" - 25° = 10° 54'3" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 839.64 }{ 113.4 } = 7.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 111 }{ 2 * sin 144° 5'57" } = 94.65 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 35.8**2 - 111**2 } }{ 2 } = 27.577 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.8**2+2 * 111**2 - 80**2 } }{ 2 } = 72.12 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 111**2 - 35.8**2 } }{ 2 } = 95.08 ; ;
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