Triangle calculator SSA

Triangle has two solutions with side c=36.43881226846 and with side c=3.87444476828

#1 Obtuse scalene triangle.

Sides: a = 22.24 b = 18.8 c = 36.43881226846

Area: T = 171.2421506699
Perimeter: p = 77.47881226846
Semiperimeter: s = 38.73990613423

Angle ∠ A = α = 29.99765870083° = 29°59'48″ = 0.52435392077 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 125.0033412992° = 125°12″ = 2.18217211329 rad

Height: h_a = 15.39994160701
Height: h_b = 18.21771815637
Height: h_c = 9.39990301411

Median: m_a = 26.77656230999
Median: m_b = 28.68547902623
Median: m_c = 9.59765933437

Inradius: r = 4.42203834777
Circumradius: R = 22.24222948816

Vertex coordinates: A[36.43881226846; 0] B[0; 0] C[20.15662851837; 9.39990301411]
Centroid: CG[18.86548026228; 3.1333010047]
Coordinates of the circumscribed circle: U[18.21990613423; -12.75987415291]
Coordinates of the inscribed circle: I[19.93990613423; 4.42203834777]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0033412992° = 150°12″ = 0.52435392077 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 54.99765870083° = 54°59'48″ = 2.18217211329 rad

How did we calculate this triangle?

1. Use Law of Cosines

$a = 22.24 ; ; b = 18.8 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 18.8**2 = 22.24**2 + c**2 -2 * 22.24 * c * cos (25° ) ; ; ; ; c**2 -40.313c +141.178 =0 ; ; p=1; q=-40.313; r=141.178 ; ; D = q**2 - 4pr = 40.313**2 - 4 * 1 * 141.178 = 1060.39292963 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 40.31 ± sqrt{ 1060.39 } }{ 2 } ; ; c_{1,2} = 20.15628518 ± 16.2818375009 ; ; c_{1} = 36.4381226809 ; ; c_{2} = 3.87444767908 ; ; ; ; text{ Factored form: } ; ; (c -36.4381226809) (c -3.87444767908) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 22.24 ; ; b = 18.8 ; ; c = 36.44 ; ;$

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

$p = a+b+c = 22.24+18.8+36.44 = 77.48 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 77.48 }{ 2 } = 38.74 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.74 * (38.74-22.24)(38.74-18.8)(38.74-36.44) } ; ; T = sqrt{ 29323.65 } = 171.24 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 171.24 }{ 22.24 } = 15.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 171.24 }{ 18.8 } = 18.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 171.24 }{ 36.44 } = 9.4 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18.8**2+36.44**2-22.24**2 }{ 2 * 18.8 * 36.44 } ) = 29° 59'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.24**2+36.44**2-18.8**2 }{ 2 * 22.24 * 36.44 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 29° 59'48" - 25° = 125° 12" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 171.24 }{ 38.74 } = 4.42 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.24 }{ 2 * sin 29° 59'48" } = 22.24 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.8**2+2 * 36.44**2 - 22.24**2 } }{ 2 } = 26.776 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 36.44**2+2 * 22.24**2 - 18.8**2 } }{ 2 } = 28.685 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.8**2+2 * 22.24**2 - 36.44**2 } }{ 2 } = 9.597 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 22.24 b = 18.8 c = 3.87444476828

Area: T = 18.20880252753
Perimeter: p = 44.91444476828
Semiperimeter: s = 22.45772238414

Angle ∠ A = α = 150.0033412992° = 150°12″ = 2.61880534459 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 4.99765870083° = 4°59'48″ = 0.08772068947 rad

Height: h_a = 1.63774123449
Height: h_b = 1.93770239655
Height: h_c = 9.39990301411

Median: m_a = 7.78327548094
Median: m_b = 12.90217236222
Median: m_c = 20.50106332534

Inradius: r = 0.81107870057
Circumradius: R = 22.24222948816

Vertex coordinates: A[3.87444476828; 0] B[0; 0] C[20.15662851837; 9.39990301411]
Centroid: CG[8.01102442888; 3.1333010047]
Coordinates of the circumscribed circle: U[1.93772238414; 22.15877716702]
Coordinates of the inscribed circle: I[3.65772238414; 0.81107870057]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99765870083° = 29°59'48″ = 2.61880534459 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 175.0033412992° = 175°12″ = 0.08772068947 rad

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How did we calculate this triangle?

1. Use Law of Cosines

$a = 22.24 ; ; b = 18.8 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 18.8**2 = 22.24**2 + c**2 -2 * 22.24 * c * cos (25° ) ; ; ; ; c**2 -40.313c +141.178 =0 ; ; p=1; q=-40.313; r=141.178 ; ; D = q**2 - 4pr = 40.313**2 - 4 * 1 * 141.178 = 1060.39292963 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 40.31 ± sqrt{ 1060.39 } }{ 2 } ; ; c_{1,2} = 20.15628518 ± 16.2818375009 ; ; c_{1} = 36.4381226809 ; ; c_{2} = 3.87444767908 ; ; ; ; text{ Factored form: } ; ; (c -36.4381226809) (c -3.87444767908) = 0 ; ; ; ; c>0 ; ; : Nr. 1$

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.

$a = 22.24 ; ; b = 18.8 ; ; c = 3.87 ; ;$

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

$p = a+b+c = 22.24+18.8+3.87 = 44.91 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 44.91 }{ 2 } = 22.46 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.46 * (22.46-22.24)(22.46-18.8)(22.46-3.87) } ; ; T = sqrt{ 331.53 } = 18.21 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.21 }{ 22.24 } = 1.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.21 }{ 18.8 } = 1.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.21 }{ 3.87 } = 9.4 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18.8**2+3.87**2-22.24**2 }{ 2 * 18.8 * 3.87 } ) = 150° 12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.24**2+3.87**2-18.8**2 }{ 2 * 22.24 * 3.87 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 150° 12" - 25° = 4° 59'48" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.21 }{ 22.46 } = 0.81 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.24 }{ 2 * sin 150° 12" } = 22.24 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.8**2+2 * 3.87**2 - 22.24**2 } }{ 2 } = 7.783 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.87**2+2 * 22.24**2 - 18.8**2 } }{ 2 } = 12.902 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.8**2+2 * 22.24**2 - 3.87**2 } }{ 2 } = 20.501 ; ;$
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