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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 111 ; ; b = 80 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 80**2 = 111**2 + c**2 -2 * 80 * c * cos (25° ) ; ; ; ; c**2 -201.2c +5921 =0 ; ; p=1; q=-201.200328722; 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} = 100.600164361 ± 64.8027242442 ; ; c_{1} = 165.402888605 ; ;
c_{2} = 35.7974401168 ; ; ; ; (c -165.402888605) (c -35.7974401168) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 111**2-80**2-35.8**2 }{ 2 * 80 * 35.8 } ) = 144° 5'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 80**2-111**2-35.8**2 }{ 2 * 111 * 35.8 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 35.8**2-111**2-80**2 }{ 2 * 80 * 111 } ) = 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 ; ;




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