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?

1. Use Law of Cosines

a = 102 ; ; b = 60 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 102**2 + c**2 -2 * 102 * c * cos (30° ) ; ; ; ; c**2 -176.669c +6804 =0 ; ; p=1; q=-176.669; 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} = 88.33459119 ± 31.6069612586 ; ; c_{1} = 119.941552449 ; ; c_{2} = 56.7276299314 ; ; ; ; text{ Factored form: } ; ; (c -119.941552449) (c -56.7276299314) = 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 = 119.94 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 60**2+119.94**2-102**2 }{ 2 * 60 * 119.94 } ) = 58° 12'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 102**2+119.94**2-60**2 }{ 2 * 102 * 119.94 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 58° 12'42" - 30° = 91° 47'18" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 119.94**2 - 102**2 } }{ 2 } = 79.95 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 119.94**2+2 * 102**2 - 60**2 } }{ 2 } = 107.215 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 102**2 - 119.94**2 } }{ 2 } = 58.357 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 102 ; ; b = 60 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 60**2 = 102**2 + c**2 -2 * 102 * c * cos (30° ) ; ; ; ; c**2 -176.669c +6804 =0 ; ; p=1; q=-176.669; 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} = 88.33459119 ± 31.6069612586 ; ; c_{1} = 119.941552449 ; ; c_{2} = 56.7276299314 ; ; ; ; text{ Factored form: } ; ; (c -119.941552449) (c -56.7276299314) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 60**2+56.73**2-102**2 }{ 2 * 60 * 56.73 } ) = 121° 47'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 102**2+56.73**2-60**2 }{ 2 * 102 * 56.73 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 121° 47'18" - 30° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 56.73**2 - 102**2 } }{ 2 } = 28.426 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.73**2+2 * 102**2 - 60**2 } }{ 2 } = 76.883 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 102**2 - 56.73**2 } }{ 2 } = 78.724 ; ;
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