Triangle calculator SSA

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Triangle has two solutions with side c=74.49548974278 and with side c=25.50551025722

#1 Acute scalene triangle.

Sides: a = 100   b = 90   c = 74.49548974278

Area: T = 3225.724368124
Perimeter: p = 264.4954897428
Semiperimeter: s = 132.2477448714

Angle ∠ A = α = 74.20768309517° = 74°12'25″ = 1.29551535276 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 45.79331690483° = 45°47'35″ = 0.79992415748 rad

Height: ha = 64.51444736248
Height: hb = 71.6832748472
Height: hc = 86.60325403784

Median: ma = 65.76327924543
Median: mb = 75.82770721536
Median: mc = 87.53664356386

Inradius: r = 24.39215758876
Circumradius: R = 51.96215242271

Vertex coordinates: A[74.49548974278; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[41.49882991426; 28.86875134595]
Coordinates of the circumscribed circle: U[37.24774487139; 36.23302023774]
Coordinates of the inscribed circle: I[42.24774487139; 24.39215758876]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.7933169048° = 105°47'35″ = 1.29551535276 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 134.2076830952° = 134°12'25″ = 0.79992415748 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 = 100 ; ; b = 90 ; ; c = 74.49 ; ;

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

p = a+b+c = 100+90+74.49 = 264.49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 264.49 }{ 2 } = 132.25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.25 * (132.25-100)(132.25-90)(132.25-74.49) } ; ; T = sqrt{ 10405293.27 } = 3225.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3225.72 }{ 100 } = 64.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3225.72 }{ 90 } = 71.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3225.72 }{ 74.49 } = 86.6 ; ;

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{ 100**2-90**2-74.49**2 }{ 2 * 90 * 74.49 } ) = 74° 12'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-74.49**2 }{ 2 * 100 * 74.49 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 74.49**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 45° 47'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3225.72 }{ 132.25 } = 24.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 74° 12'25" } = 51.96 ; ;





#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 25.50551025722

Area: T = 1104.403333768
Perimeter: p = 215.5055102572
Semiperimeter: s = 107.7532551286

Angle ∠ A = α = 105.7933169048° = 105°47'35″ = 1.8466439126 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 14.20768309517° = 14°12'25″ = 0.24879559764 rad

Height: ha = 22.08880667536
Height: hb = 24.54222963929
Height: hc = 86.60325403784

Median: ma = 43.30442160604
Median: mb = 57.4487847032
Median: mc = 94.27328616077

Inradius: r = 10.24994402638
Circumradius: R = 51.96215242271

Vertex coordinates: A[25.50551025722; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[25.16883675241; 28.86875134595]
Coordinates of the circumscribed circle: U[12.75325512861; 50.37223380011]
Coordinates of the inscribed circle: I[17.75325512861; 10.24994402638]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 74.20768309517° = 74°12'25″ = 1.8466439126 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 165.7933169048° = 165°47'35″ = 0.24879559764 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 100**2 + c**2 -2 * 90 * c * cos (60° ) ; ; ; ; c**2 -100c +1900 =0 ; ; p=1; q=-100; r=1900 ; ; D = q**2 - 4pr = 100**2 - 4 * 1 * 1900 = 2400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 100 ± sqrt{ 2400 } }{ 2 } = 50 ± sqrt{ 600 } ; ; c_{1,2} = 50 ± 24.4948974278 ; ; c_{1} = 74.4948974278 ; ;
c_{2} = 25.5051025722 ; ; ; ; (c -74.4948974278) (c -25.5051025722) = 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 = 100 ; ; b = 90 ; ; c = 25.51 ; ;

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

p = a+b+c = 100+90+25.51 = 215.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 215.51 }{ 2 } = 107.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 107.75 * (107.75-100)(107.75-90)(107.75-25.51) } ; ; T = sqrt{ 1219706.73 } = 1104.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1104.4 }{ 100 } = 22.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1104.4 }{ 90 } = 24.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1104.4 }{ 25.51 } = 86.6 ; ;

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{ 100**2-90**2-25.51**2 }{ 2 * 90 * 25.51 } ) = 105° 47'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-100**2-25.51**2 }{ 2 * 100 * 25.51 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25.51**2-100**2-90**2 }{ 2 * 90 * 100 } ) = 14° 12'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1104.4 }{ 107.75 } = 10.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 105° 47'35" } = 51.96 ; ;




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