Triangle calculator SSA

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Triangle has two solutions with side c=59.77992384739 and with side c=22.88441991789

#1 Acute scalene triangle.

Sides: a = 63   b = 51   c = 59.77992384739

Area: T = 1421.15328652
Perimeter: p = 173.7799238474
Semiperimeter: s = 86.89896192369

Angle ∠ A = α = 68.79443766898° = 68°47'40″ = 1.20106883801 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 62.20656233102° = 62°12'20″ = 1.086569294 rad

Height: ha = 45.11659639747
Height: hb = 55.73114849099
Height: hc = 47.5476703554

Median: ma = 45.77114832211
Median: mb = 55.86661675458
Median: mc = 48.90440965755

Inradius: r = 16.35658417873
Circumradius: R = 33.78878313304

Vertex coordinates: A[59.77992384739; 0] B[0; 0] C[41.33217188264; 47.5476703554]
Centroid: CG[33.70436524334; 15.84989011847]
Coordinates of the circumscribed circle: U[29.89896192369; 15.75552596894]
Coordinates of the inscribed circle: I[35.89896192369; 16.35658417873]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.206562331° = 111°12'20″ = 1.20106883801 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 117.794437669° = 117°47'40″ = 1.086569294 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 = 63 ; ; b = 51 ; ; c = 59.78 ; ;

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

p = a+b+c = 63+51+59.78 = 173.78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 173.78 }{ 2 } = 86.89 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.89 * (86.89-63)(86.89-51)(86.89-59.78) } ; ; T = sqrt{ 2019675.47 } = 1421.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1421.15 }{ 63 } = 45.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1421.15 }{ 51 } = 55.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1421.15 }{ 59.78 } = 47.55 ; ;

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{ 63**2-51**2-59.78**2 }{ 2 * 51 * 59.78 } ) = 68° 47'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 51**2-63**2-59.78**2 }{ 2 * 63 * 59.78 } ) = 49° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 59.78**2-63**2-51**2 }{ 2 * 51 * 63 } ) = 62° 12'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1421.15 }{ 86.89 } = 16.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 63 }{ 2 * sin 68° 47'40" } = 33.79 ; ;





#2 Obtuse scalene triangle.

Sides: a = 63   b = 51   c = 22.88441991789

Area: T = 544.0344117216
Perimeter: p = 136.8844199179
Semiperimeter: s = 68.44220995895

Angle ∠ A = α = 111.206562331° = 111°12'20″ = 1.94109042735 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 19.79443766898° = 19°47'40″ = 0.34554770466 rad

Height: ha = 17.2710924356
Height: hb = 21.33546712634
Height: hc = 47.5476703554

Median: ma = 23.87766263536
Median: mb = 39.95111362295
Median: mc = 56.16111819408

Inradius: r = 7.94988227345
Circumradius: R = 33.78878313304

Vertex coordinates: A[22.88441991789; 0] B[0; 0] C[41.33217188264; 47.5476703554]
Centroid: CG[21.40553060018; 15.84989011847]
Coordinates of the circumscribed circle: U[11.44220995895; 31.79114438646]
Coordinates of the inscribed circle: I[17.44220995895; 7.94988227345]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 68.79443766898° = 68°47'40″ = 1.94109042735 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 160.206562331° = 160°12'20″ = 0.34554770466 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 63 ; ; b = 51 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 51**2 = 63**2 + c**2 -2 * 51 * c * cos (49° ) ; ; ; ; c**2 -82.663c +1368 =0 ; ; p=1; q=-82.6634376528; r=1368 ; ; D = q**2 - 4pr = 82.663**2 - 4 * 1 * 1368 = 1361.24392458 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 82.66 ± sqrt{ 1361.24 } }{ 2 } ; ; c_{1,2} = 41.3317188264 ± 18.4475196475 ; ; c_{1} = 59.7792384739 ; ;
c_{2} = 22.8841991789 ; ; ; ; (c -59.7792384739) (c -22.8841991789) = 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 = 63 ; ; b = 51 ; ; c = 22.88 ; ;

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

p = a+b+c = 63+51+22.88 = 136.88 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 136.88 }{ 2 } = 68.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 68.44 * (68.44-63)(68.44-51)(68.44-22.88) } ; ; T = sqrt{ 295973.12 } = 544.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 544.03 }{ 63 } = 17.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 544.03 }{ 51 } = 21.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 544.03 }{ 22.88 } = 47.55 ; ;

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{ 63**2-51**2-22.88**2 }{ 2 * 51 * 22.88 } ) = 111° 12'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 51**2-63**2-22.88**2 }{ 2 * 63 * 22.88 } ) = 49° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.88**2-63**2-51**2 }{ 2 * 51 * 63 } ) = 19° 47'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 544.03 }{ 68.44 } = 7.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 63 }{ 2 * sin 111° 12'20" } = 33.79 ; ;




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