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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 130**2 + c**2 -2 * 90 * c * cos (30° ) ; ; ; ; c**2 -225.167c +8800 =0 ; ; p=1; q=-225.166604984; 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} = 112.583302492 ± 62.2494979899 ; ; c_{1} = 174.832800482 ; ;
c_{2} = 50.333804502 ; ; ; ; (c -174.832800482) (c -50.333804502) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 130**2-90**2-50.33**2 }{ 2 * 90 * 50.33 } ) = 133° 45'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-50.33**2 }{ 2 * 130 * 50.33 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 50.33**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 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 ; ;




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