Triangle calculator SSA

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Triangle has two solutions with side c=48.76438049792 and with side c=5.90660198465

#1 Acute scalene triangle.

Sides: a = 38   b = 34   c = 48.76438049792

Area: T = 643.610952078
Perimeter: p = 120.7643804979
Semiperimeter: s = 60.38219024896

Angle ∠ A = α = 50.93105784325° = 50°55'50″ = 0.88989062836 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 85.06994215675° = 85°4'10″ = 1.48547414991 rad

Height: ha = 33.87441853042
Height: hb = 37.85993835753
Height: hc = 26.39770180774

Median: ma = 37.49660576331
Median: mb = 40.27334942366
Median: mc = 26.56216797471

Inradius: r = 10.65989804932
Circumradius: R = 24.47224611736

Vertex coordinates: A[48.76438049792; 0] B[0; 0] C[27.33549124129; 26.39770180774]
Centroid: CG[25.36662391307; 8.79990060258]
Coordinates of the circumscribed circle: U[24.38219024896; 2.1033375117]
Coordinates of the inscribed circle: I[26.38219024896; 10.65989804932]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.0699421568° = 129°4'10″ = 0.88989062836 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 94.93105784325° = 94°55'50″ = 1.48547414991 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 = 38 ; ; b = 34 ; ; c = 48.76 ; ;

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

p = a+b+c = 38+34+48.76 = 120.76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 120.76 }{ 2 } = 60.38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.38 * (60.38-38)(60.38-34)(60.38-48.76) } ; ; T = sqrt{ 414233.22 } = 643.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 643.61 }{ 38 } = 33.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 643.61 }{ 34 } = 37.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 643.61 }{ 48.76 } = 26.4 ; ;

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{ 38**2-34**2-48.76**2 }{ 2 * 34 * 48.76 } ) = 50° 55'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-38**2-48.76**2 }{ 2 * 38 * 48.76 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.76**2-38**2-34**2 }{ 2 * 34 * 38 } ) = 85° 4'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 643.61 }{ 60.38 } = 10.66 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38 }{ 2 * sin 50° 55'50" } = 24.47 ; ;





#2 Obtuse scalene triangle.

Sides: a = 38   b = 34   c = 5.90660198465

Area: T = 77.95106563274
Perimeter: p = 77.90660198465
Semiperimeter: s = 38.95330099233

Angle ∠ A = α = 129.0699421568° = 129°4'10″ = 2.253268637 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 6.93105784325° = 6°55'50″ = 0.12109614127 rad

Height: ha = 4.10326661225
Height: hb = 4.58553327251
Height: hc = 26.39770180774

Median: ma = 15.31114511139
Median: mb = 21.22435844101
Median: mc = 35.9344380924

Inradius: r = 2.00111459058
Circumradius: R = 24.47224611736

Vertex coordinates: A[5.90660198465; 0] B[0; 0] C[27.33549124129; 26.39770180774]
Centroid: CG[11.08803107531; 8.79990060258]
Coordinates of the circumscribed circle: U[2.95330099233; 24.29436429604]
Coordinates of the inscribed circle: I[4.95330099233; 2.00111459058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 50.93105784325° = 50°55'50″ = 2.253268637 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 173.0699421568° = 173°4'10″ = 0.12109614127 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 38 ; ; b = 34 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 34**2 = 38**2 + c**2 -2 * 34 * c * cos (44° ) ; ; ; ; c**2 -54.67c +288 =0 ; ; p=1; q=-54.6698248257; r=288 ; ; D = q**2 - 4pr = 54.67**2 - 4 * 1 * 288 = 1836.78974648 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 54.67 ± sqrt{ 1836.79 } }{ 2 } ; ; c_{1,2} = 27.3349124129 ± 21.4288925663 ; ; c_{1} = 48.7638049792 ; ;
c_{2} = 5.90601984654 ; ; ; ; (c -48.7638049792) (c -5.90601984654) = 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 = 38 ; ; b = 34 ; ; c = 5.91 ; ;

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

p = a+b+c = 38+34+5.91 = 77.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77.91 }{ 2 } = 38.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.95 * (38.95-38)(38.95-34)(38.95-5.91) } ; ; T = sqrt{ 6076.3 } = 77.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.95 }{ 38 } = 4.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.95 }{ 34 } = 4.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.95 }{ 5.91 } = 26.4 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 38**2-34**2-5.91**2 }{ 2 * 34 * 5.91 } ) = 129° 4'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-38**2-5.91**2 }{ 2 * 38 * 5.91 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.91**2-38**2-34**2 }{ 2 * 34 * 38 } ) = 6° 55'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.95 }{ 38.95 } = 2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38 }{ 2 * sin 129° 4'10" } = 24.47 ; ;




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