Triangle calculator SSA

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Triangle has two solutions with side c=98.43107429352 and with side c=17.27110268083

#1 Acute scalene triangle.

Sides: a = 90   b = 80   c = 98.43107429352

Area: T = 3393.105456459
Perimeter: p = 268.4310742935
Semiperimeter: s = 134.2155371468

Angle ∠ A = α = 59.51992913471° = 59°31'9″ = 1.03988076025 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 70.48107086529° = 70°28'51″ = 1.23301204251 rad

Height: ha = 75.40223236576
Height: hb = 84.82876141148
Height: hc = 68.94439998807

Median: ma = 77.58441838095
Median: mb = 85.40767068642
Median: mc = 69.48327116001

Inradius: r = 25.28110429051
Circumradius: R = 52.21662915733

Vertex coordinates: A[98.43107429352; 0] B[0; 0] C[57.85108848718; 68.94439998807]
Centroid: CG[52.09438759357; 22.98113332936]
Coordinates of the circumscribed circle: U[49.21553714676; 17.44767279732]
Coordinates of the inscribed circle: I[54.21553714676; 25.28110429051]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.4810708653° = 120°28'51″ = 1.03988076025 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 109.5199291347° = 109°31'9″ = 1.23301204251 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 90 ; ; b = 80 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 80**2 = 90**2 + c**2 -2 * 90 * c * cos (50° ) ; ; ; ; c**2 -115.702c +1700 =0 ; ; p=1; q=-115.702; r=1700 ; ; D = q**2 - 4pr = 115.702**2 - 4 * 1 * 1700 = 6586.8995218 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 115.7 ± sqrt{ 6586.9 } }{ 2 } ; ; c_{1,2} = 57.85088487 ± 40.5798580634 ; ; c_{1} = 98.4307429334 ; ; c_{2} = 17.2710268066 ; ; ; ; text{ Factored form: } ; ; (c -98.4307429334) (c -17.2710268066) = 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 = 90 ; ; b = 80 ; ; c = 98.43 ; ;

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

p = a+b+c = 90+80+98.43 = 268.43 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 268.43 }{ 2 } = 134.22 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.22 * (134.22-90)(134.22-80)(134.22-98.43) } ; ; T = sqrt{ 11513158.59 } = 3393.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3393.1 }{ 90 } = 75.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3393.1 }{ 80 } = 84.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3393.1 }{ 98.43 } = 68.94 ; ;

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+98.43**2-90**2 }{ 2 * 80 * 98.43 } ) = 59° 31'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+98.43**2-80**2 }{ 2 * 90 * 98.43 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 59° 31'9" - 50° = 70° 28'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3393.1 }{ 134.22 } = 25.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 59° 31'9" } = 52.22 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 98.43**2 - 90**2 } }{ 2 } = 77.584 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 98.43**2+2 * 90**2 - 80**2 } }{ 2 } = 85.407 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 90**2 - 98.43**2 } }{ 2 } = 69.483 ; ;







#2 Obtuse scalene triangle.

Sides: a = 90   b = 80   c = 17.27110268083

Area: T = 595.3676835107
Perimeter: p = 187.2711026808
Semiperimeter: s = 93.63655134042

Angle ∠ A = α = 120.4810708653° = 120°28'51″ = 2.10327850511 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 9.51992913471° = 9°31'9″ = 0.16661429765 rad

Height: ha = 13.23303741135
Height: hb = 14.88441708777
Height: hc = 68.94439998807

Median: ma = 36.3898792004
Median: mb = 50.98218024741
Median: mc = 84.70878975553

Inradius: r = 6.35883443232
Circumradius: R = 52.21662915733

Vertex coordinates: A[17.27110268083; 0] B[0; 0] C[57.85108848718; 68.94439998807]
Centroid: CG[25.04106372267; 22.98113332936]
Coordinates of the circumscribed circle: U[8.63655134042; 51.49772719075]
Coordinates of the inscribed circle: I[13.63655134042; 6.35883443232]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 59.51992913471° = 59°31'9″ = 2.10327850511 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 170.4810708653° = 170°28'51″ = 0.16661429765 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 90 ; ; b = 80 ; ; beta = 50° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 80**2 = 90**2 + c**2 -2 * 90 * c * cos (50° ) ; ; ; ; c**2 -115.702c +1700 =0 ; ; p=1; q=-115.702; r=1700 ; ; D = q**2 - 4pr = 115.702**2 - 4 * 1 * 1700 = 6586.8995218 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 115.7 ± sqrt{ 6586.9 } }{ 2 } ; ; c_{1,2} = 57.85088487 ± 40.5798580634 ; ; c_{1} = 98.4307429334 ; ; c_{2} = 17.2710268066 ; ; ; ; text{ Factored form: } ; ; (c -98.4307429334) (c -17.2710268066) = 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 = 90 ; ; b = 80 ; ; c = 17.27 ; ;

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

p = a+b+c = 90+80+17.27 = 187.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 187.27 }{ 2 } = 93.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 93.64 * (93.64-90)(93.64-80)(93.64-17.27) } ; ; T = sqrt{ 354461.67 } = 595.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 595.37 }{ 90 } = 13.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 595.37 }{ 80 } = 14.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 595.37 }{ 17.27 } = 68.94 ; ;

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+17.27**2-90**2 }{ 2 * 80 * 17.27 } ) = 120° 28'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+17.27**2-80**2 }{ 2 * 90 * 17.27 } ) = 50° ; ; gamma = 180° - alpha - beta = 180° - 120° 28'51" - 50° = 9° 31'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 595.37 }{ 93.64 } = 6.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 120° 28'51" } = 52.22 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 17.27**2 - 90**2 } }{ 2 } = 36.389 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.27**2+2 * 90**2 - 80**2 } }{ 2 } = 50.982 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 90**2 - 17.27**2 } }{ 2 } = 84.708 ; ;
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