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?

1. Use Law of Cosines

a = 63 ; ; b = 51 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 51**2 = 63**2 + c**2 -2 * 63 * c * cos (49° ) ; ; ; ; c**2 -82.663c +1368 =0 ; ; p=1; q=-82.663; 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} = 41.33171883 ± 18.4475196475 ; ; c_{1} = 59.7792384775 ; ; c_{2} = 22.8841991825 ; ; ; ; text{ Factored form: } ; ; (c -59.7792384775) (c -22.8841991825) = 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 = 59.78 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 51**2+59.78**2-63**2 }{ 2 * 51 * 59.78 } ) = 68° 47'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63**2+59.78**2-51**2 }{ 2 * 63 * 59.78 } ) = 49° ; ; gamma = 180° - alpha - beta = 180° - 68° 47'40" - 49° = 62° 12'20" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 59.78**2 - 63**2 } }{ 2 } = 45.771 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.78**2+2 * 63**2 - 51**2 } }{ 2 } = 55.866 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 63**2 - 59.78**2 } }{ 2 } = 48.904 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 63 ; ; b = 51 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 51**2 = 63**2 + c**2 -2 * 63 * c * cos (49° ) ; ; ; ; c**2 -82.663c +1368 =0 ; ; p=1; q=-82.663; 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} = 41.33171883 ± 18.4475196475 ; ; c_{1} = 59.7792384775 ; ; c_{2} = 22.8841991825 ; ; ; ; text{ Factored form: } ; ; (c -59.7792384775) (c -22.8841991825) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 51**2+22.88**2-63**2 }{ 2 * 51 * 22.88 } ) = 111° 12'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63**2+22.88**2-51**2 }{ 2 * 63 * 22.88 } ) = 49° ; ; gamma = 180° - alpha - beta = 180° - 111° 12'20" - 49° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 22.88**2 - 63**2 } }{ 2 } = 23.877 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.88**2+2 * 63**2 - 51**2 } }{ 2 } = 39.951 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 63**2 - 22.88**2 } }{ 2 } = 56.161 ; ;
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