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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 77**2 = 107**2 + c**2 -2 * 77 * c * cos (28° ) ; ; ; ; c**2 -188.951c +5520 =0 ; ; p=1; q=-188.950784872; 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} = 94.4753924359 ± 58.3575168759 ; ; c_{1} = 152.832909312 ; ;
c_{2} = 36.11787556 ; ; ; ; (c -152.832909312) (c -36.11787556) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 107**2-77**2-36.12**2 }{ 2 * 77 * 36.12 } ) = 139° 16'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 77**2-107**2-36.12**2 }{ 2 * 107 * 36.12 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 36.12**2-107**2-77**2 }{ 2 * 77 * 107 } ) = 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 ; ;




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