Triangle calculator SSA

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Triangle has two solutions with side c=174.8332800482 and with side c=50.3343804502

#1 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 174.8332800482

Area: T = 5682.066601566
Perimeter: p = 394.8332800482
Semiperimeter: s = 197.4166400241

Angle ∠ A = α = 46.23882573073° = 46°14'18″ = 0.80770098304 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 103.7621742693° = 103°45'42″ = 1.81109840476 rad

Height: ha = 87.4166400241
Height: hb = 126.2688133681
Height: hc = 65

Median: ma = 122.9165637989
Median: mb = 147.3377212075
Median: mc = 69.70220298766

Inradius: r = 28.78221376984
Circumradius: R = 90

Vertex coordinates: A[174.8332800482; 0] B[0; 0] C[112.5833302492; 65]
Centroid: CG[95.8055367658; 21.66766666667]
Coordinates of the circumscribed circle: U[87.4166400241; -21.41096466321]
Coordinates of the inscribed circle: I[107.4166400241; 28.78221376984]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.7621742693° = 133°45'42″ = 0.80770098304 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 76.23882573073° = 76°14'18″ = 1.81109840476 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (30° ) ; ; ; ; c**2 -225.167c +8800 =0 ; ; p=1; q=-225.167; r=8800 ; ; D = q**2 - 4pr = 225.167**2 - 4 * 1 * 8800 = 15500 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 225.17 ± sqrt{ 15500 } }{ 2 } ; ; c_{1,2} = 112.58330249 ± 62.2494979899 ; ; c_{1} = 174.83280048 ; ;
c_{2} = 50.3338045001 ; ; ; ; text{ Factored form: } ; ; (c -174.83280048) (c -50.3338045001) = 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 = 130 ; ; b = 90 ; ; c = 174.83 ; ;

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

p = a+b+c = 130+90+174.83 = 394.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 394.83 }{ 2 } = 197.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 197.42 * (197.42-130)(197.42-90)(197.42-174.83) } ; ; T = sqrt{ 32285874.21 } = 5682.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5682.07 }{ 130 } = 87.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5682.07 }{ 90 } = 126.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5682.07 }{ 174.83 } = 65 ; ;

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{ 90**2+174.83**2-130**2 }{ 2 * 90 * 174.83 } ) = 46° 14'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 130**2+174.83**2-90**2 }{ 2 * 130 * 174.83 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 46° 14'18" - 30° = 103° 45'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5682.07 }{ 197.42 } = 28.78 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 130 }{ 2 * sin 46° 14'18" } = 90 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 174.83**2 - 130**2 } }{ 2 } = 122.916 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 174.83**2+2 * 130**2 - 90**2 } }{ 2 } = 147.337 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 130**2 - 174.83**2 } }{ 2 } = 69.702 ; ;







#2 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 50.3343804502

Area: T = 1635.849864632
Perimeter: p = 270.3343804502
Semiperimeter: s = 135.1676902251

Angle ∠ A = α = 133.7621742693° = 133°45'42″ = 2.33545828232 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 16.23882573073° = 16°14'18″ = 0.28334110548 rad

Height: ha = 25.1676902251
Height: hb = 36.35221921404
Height: hc = 65

Median: ma = 33.04215789245
Median: mb = 87.70325993789
Median: mc = 108.9344048998

Inradius: r = 12.10224349828
Circumradius: R = 90

Vertex coordinates: A[50.3343804502; 0] B[0; 0] C[112.5833302492; 65]
Centroid: CG[54.30657023313; 21.66766666667]
Coordinates of the circumscribed circle: U[25.1676902251; 86.41096466321]
Coordinates of the inscribed circle: I[45.1676902251; 12.10224349828]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 46.23882573073° = 46°14'18″ = 2.33545828232 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 163.7621742693° = 163°45'42″ = 0.28334110548 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (30° ) ; ; ; ; c**2 -225.167c +8800 =0 ; ; p=1; q=-225.167; r=8800 ; ; D = q**2 - 4pr = 225.167**2 - 4 * 1 * 8800 = 15500 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 225.17 ± sqrt{ 15500 } }{ 2 } ; ; c_{1,2} = 112.58330249 ± 62.2494979899 ; ; c_{1} = 174.83280048 ; ; : Nr. 1
c_{2} = 50.3338045001 ; ; ; ; text{ Factored form: } ; ; (c -174.83280048) (c -50.3338045001) = 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 = 130 ; ; b = 90 ; ; c = 50.33 ; ;

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

p = a+b+c = 130+90+50.33 = 270.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 270.33 }{ 2 } = 135.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.17 * (135.17-130)(135.17-90)(135.17-50.33) } ; ; T = sqrt{ 2676000.79 } = 1635.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1635.85 }{ 130 } = 25.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1635.85 }{ 90 } = 36.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1635.85 }{ 50.33 } = 65 ; ;

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{ 90**2+50.33**2-130**2 }{ 2 * 90 * 50.33 } ) = 133° 45'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 130**2+50.33**2-90**2 }{ 2 * 130 * 50.33 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 133° 45'42" - 30° = 16° 14'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1635.85 }{ 135.17 } = 12.1 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 130 }{ 2 * sin 133° 45'42" } = 90 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 50.33**2 - 130**2 } }{ 2 } = 33.042 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 50.33**2+2 * 130**2 - 90**2 } }{ 2 } = 87.703 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 130**2 - 50.33**2 } }{ 2 } = 108.934 ; ;
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