Triangle calculator SSA

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Triangle has two solutions with side c=119.9421552445 and with side c=56.72876299275

#1 Obtuse scalene triangle.

Sides: a = 102   b = 60   c = 119.9421552445

Area: T = 3058.510958734
Perimeter: p = 281.9421552445
Semiperimeter: s = 140.9710776222

Angle ∠ A = α = 58.21216693829° = 58°12'42″ = 1.01659852938 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 91.78883306171° = 91°47'18″ = 1.60220085842 rad

Height: ha = 59.97107762223
Height: hb = 101.9550319578
Height: hc = 51

Median: ma = 79.95499093271
Median: mb = 107.2154681837
Median: mc = 58.35767134038

Inradius: r = 21.69660540993
Circumradius: R = 60

Vertex coordinates: A[119.9421552445; 0] B[0; 0] C[88.3354591186; 51]
Centroid: CG[69.42553812102; 17]
Coordinates of the circumscribed circle: U[59.97107762223; -1.87224313863]
Coordinates of the inscribed circle: I[80.97107762223; 21.69660540993]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.7888330617° = 121°47'18″ = 1.01659852938 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 88.21216693829° = 88°12'42″ = 1.60220085842 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 = 102 ; ; b = 60 ; ; c = 119.94 ; ;

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

p = a+b+c = 102+60+119.94 = 281.94 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 281.94 }{ 2 } = 140.97 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 140.97 * (140.97-102)(140.97-60)(140.97-119.94) } ; ; T = sqrt{ 9354480.9 } = 3058.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3058.51 }{ 102 } = 59.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3058.51 }{ 60 } = 101.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3058.51 }{ 119.94 } = 51 ; ;

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{ 102**2-60**2-119.94**2 }{ 2 * 60 * 119.94 } ) = 58° 12'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-102**2-119.94**2 }{ 2 * 102 * 119.94 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 119.94**2-102**2-60**2 }{ 2 * 60 * 102 } ) = 91° 47'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3058.51 }{ 140.97 } = 21.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 102 }{ 2 * sin 58° 12'42" } = 60 ; ;





#2 Obtuse scalene triangle.

Sides: a = 102   b = 60   c = 56.72876299275

Area: T = 1446.555456315
Perimeter: p = 218.7287629928
Semiperimeter: s = 109.3643814964

Angle ∠ A = α = 121.7888330617° = 121°47'18″ = 2.12656073598 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 28.21216693829° = 28°12'42″ = 0.49223865182 rad

Height: ha = 28.36438149637
Height: hb = 48.21884854383
Height: hc = 51

Median: ma = 28.42655518608
Median: mb = 76.88331060675
Median: mc = 78.72441640204

Inradius: r = 13.22769943549
Circumradius: R = 60

Vertex coordinates: A[56.72876299275; 0] B[0; 0] C[88.3354591186; 51]
Centroid: CG[48.35440737045; 17]
Coordinates of the circumscribed circle: U[28.36438149637; 52.87224313863]
Coordinates of the inscribed circle: I[49.36438149637; 13.22769943549]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.21216693829° = 58°12'42″ = 2.12656073598 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 151.7888330617° = 151°47'18″ = 0.49223865182 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 102 ; ; b = 60 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 60**2 = 102**2 + c**2 -2 * 60 * c * cos (30° ) ; ; ; ; c**2 -176.669c +6804 =0 ; ; p=1; q=-176.669182372; r=6804 ; ; D = q**2 - 4pr = 176.669**2 - 4 * 1 * 6804 = 3996 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.67 ± sqrt{ 3996 } }{ 2 } = fraction{ 176.67 ± 6 sqrt{ 111 } }{ 2 } ; ; c_{1,2} = 88.334591186 ± 31.6069612586 ; ;
c_{1} = 119.941552445 ; ; c_{2} = 56.7276299275 ; ; ; ; (c -119.941552445) (c -56.7276299275) = 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 = 102 ; ; b = 60 ; ; c = 56.73 ; ;

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

p = a+b+c = 102+60+56.73 = 218.73 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 218.73 }{ 2 } = 109.36 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 109.36 * (109.36-102)(109.36-60)(109.36-56.73) } ; ; T = sqrt{ 2092520.1 } = 1446.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1446.55 }{ 102 } = 28.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1446.55 }{ 60 } = 48.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1446.55 }{ 56.73 } = 51 ; ;

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{ 102**2-60**2-56.73**2 }{ 2 * 60 * 56.73 } ) = 121° 47'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-102**2-56.73**2 }{ 2 * 102 * 56.73 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56.73**2-102**2-60**2 }{ 2 * 60 * 102 } ) = 28° 12'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1446.55 }{ 109.36 } = 13.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 102 }{ 2 * sin 121° 47'18" } = 60 ; ;




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