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?

1. Use Law of Cosines

a = 38 ; ; b = 34 ; ; beta = 44° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 38**2 + c**2 -2 * 38 * c * cos (44° ) ; ; ; ; c**2 -54.67c +288 =0 ; ; p=1; q=-54.67; 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} = 27.33491241 ± 21.4288925663 ; ; c_{1} = 48.7638049763 ; ;
c_{2} = 5.90601984367 ; ; ; ; (c -48.7638049763) (c -5.90601984367) = 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 = 48.76 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-48.76**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 34**2 = 38**2 + c**2 -2 * 38 * c * cos (44° ) ; ; ; ; c**2 -54.67c +288 =0 ; ; p=1; q=-54.67; 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} = 27.33491241 ± 21.4288925663 ; ; c_{1} = 48.7638049763 ; ; : Nr. 1
c_{2} = 5.90601984367 ; ; ; ; (c -48.7638049763) (c -5.90601984367) = 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 = 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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