Triangle calculator SSA

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Triangle has two solutions with side c=152.8332909312 and with side c=36.118787556

#1 Obtuse scalene triangle.

Sides: a = 107   b = 77   c = 152.8332909312

Area: T = 3838.663270571
Perimeter: p = 336.8332909312
Semiperimeter: s = 168.4166454656

Angle ∠ A = α = 40.72114785785° = 40°43'17″ = 0.71107238775 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 111.2798521421° = 111°16'43″ = 1.94221765856 rad

Height: ha = 71.75107047797
Height: hb = 99.70655248238
Height: hc = 50.23334572181

Median: ma = 108.5411232185
Median: mb = 126.1799233966
Median: mc = 53.38109465804

Inradius: r = 22.79326820664
Circumradius: R = 82.00770970253

Vertex coordinates: A[152.8332909312; 0] B[0; 0] C[94.47553924359; 50.23334572181]
Centroid: CG[82.43661005826; 16.74444857394]
Coordinates of the circumscribed circle: U[76.41664546559; -29.76105346111]
Coordinates of the inscribed circle: I[91.41664546559; 22.79326820664]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.2798521421° = 139°16'43″ = 0.71107238775 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 68.72114785785° = 68°43'17″ = 1.94221765856 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 107 ; ; b = 77 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 77**2 = 107**2 + c**2 -2 * 107 * c * cos (28° ) ; ; ; ; c**2 -188.951c +5520 =0 ; ; p=1; q=-188.951; r=5520 ; ; D = q**2 - 4pr = 188.951**2 - 4 * 1 * 5520 = 13622.3991037 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 188.95 ± sqrt{ 13622.4 } }{ 2 } ; ; c_{1,2} = 94.47539244 ± 58.3575168759 ; ; c_{1} = 152.832909316 ; ;
c_{2} = 36.1178755641 ; ; ; ; text{ Factored form: } ; ; (c -152.832909316) (c -36.1178755641) = 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 = 107 ; ; b = 77 ; ; c = 152.83 ; ;

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

p = a+b+c = 107+77+152.83 = 336.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 336.83 }{ 2 } = 168.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 168.42 * (168.42-107)(168.42-77)(168.42-152.83) } ; ; T = sqrt{ 14735331.37 } = 3838.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3838.66 }{ 107 } = 71.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3838.66 }{ 77 } = 99.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3838.66 }{ 152.83 } = 50.23 ; ;

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{ 77**2+152.83**2-107**2 }{ 2 * 77 * 152.83 } ) = 40° 43'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 107**2+152.83**2-77**2 }{ 2 * 107 * 152.83 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 40° 43'17" - 28° = 111° 16'43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3838.66 }{ 168.42 } = 22.79 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 107 }{ 2 * sin 40° 43'17" } = 82.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 152.83**2 - 107**2 } }{ 2 } = 108.541 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 152.83**2+2 * 107**2 - 77**2 } }{ 2 } = 126.179 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 107**2 - 152.83**2 } }{ 2 } = 53.381 ; ;







#2 Obtuse scalene triangle.

Sides: a = 107   b = 77   c = 36.118787556

Area: T = 907.1632878376
Perimeter: p = 220.118787556
Semiperimeter: s = 110.059893778

Angle ∠ A = α = 139.2798521421° = 139°16'43″ = 2.43108687761 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 12.72114785785° = 12°43'17″ = 0.22220316869 rad

Height: ha = 16.95663154837
Height: hb = 23.56326721656
Height: hc = 50.23334572181

Median: ma = 27.46881718992
Median: mb = 69.96107065965
Median: mc = 91.44987548644

Inradius: r = 8.24325189328
Circumradius: R = 82.00770970253

Vertex coordinates: A[36.118787556; 0] B[0; 0] C[94.47553924359; 50.23334572181]
Centroid: CG[43.5311089332; 16.74444857394]
Coordinates of the circumscribed circle: U[18.059893778; 79.99439918292]
Coordinates of the inscribed circle: I[33.059893778; 8.24325189328]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 40.72114785785° = 40°43'17″ = 2.43108687761 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 167.2798521421° = 167°16'43″ = 0.22220316869 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 107 ; ; b = 77 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 77**2 = 107**2 + c**2 -2 * 107 * c * cos (28° ) ; ; ; ; c**2 -188.951c +5520 =0 ; ; p=1; q=-188.951; r=5520 ; ; D = q**2 - 4pr = 188.951**2 - 4 * 1 * 5520 = 13622.3991037 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 188.95 ± sqrt{ 13622.4 } }{ 2 } ; ; c_{1,2} = 94.47539244 ± 58.3575168759 ; ; c_{1} = 152.832909316 ; ; : Nr. 1
c_{2} = 36.1178755641 ; ; ; ; text{ Factored form: } ; ; (c -152.832909316) (c -36.1178755641) = 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 = 107 ; ; b = 77 ; ; c = 36.12 ; ;

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

p = a+b+c = 107+77+36.12 = 220.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 220.12 }{ 2 } = 110.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.06 * (110.06-107)(110.06-77)(110.06-36.12) } ; ; T = sqrt{ 822944.49 } = 907.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 907.16 }{ 107 } = 16.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 907.16 }{ 77 } = 23.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 907.16 }{ 36.12 } = 50.23 ; ;

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{ 77**2+36.12**2-107**2 }{ 2 * 77 * 36.12 } ) = 139° 16'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 107**2+36.12**2-77**2 }{ 2 * 107 * 36.12 } ) = 28° ; ; gamma = 180° - alpha - beta = 180° - 139° 16'43" - 28° = 12° 43'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 907.16 }{ 110.06 } = 8.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 107 }{ 2 * sin 139° 16'43" } = 82.01 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 36.12**2 - 107**2 } }{ 2 } = 27.468 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 36.12**2+2 * 107**2 - 77**2 } }{ 2 } = 69.961 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77**2+2 * 107**2 - 36.12**2 } }{ 2 } = 91.449 ; ;
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