Triangle calculator SSA

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Triangle has two solutions with side c=104.3166428443 and with side c=54.43301610471

#1 Obtuse scalene triangle.

Sides: a = 83   b = 34.8   c = 104.3166428443

Area: T = 1265.71656386
Perimeter: p = 222.1166428443
Semiperimeter: s = 111.0588214221

Angle ∠ A = α = 44.21326460331° = 44°12'46″ = 0.77216562443 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 118.7877353967° = 118°47'14″ = 2.07332304365 rad

Height: ha = 30.49991720144
Height: hb = 72.74222780803
Height: hc = 24.2676851492

Median: ma = 65.75988672463
Median: mb = 92.64328552103
Median: mc = 36.46328672657

Inradius: r = 11.39768664765
Circumradius: R = 59.51332829851

Vertex coordinates: A[104.3166428443; 0] B[0; 0] C[79.37332947449; 24.2676851492]
Centroid: CG[61.23299077292; 8.08989504973]
Coordinates of the circumscribed circle: U[52.15882142214; -28.65992313383]
Coordinates of the inscribed circle: I[76.25882142214; 11.39768664765]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.7877353967° = 135°47'14″ = 0.77216562443 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 61.21326460331° = 61°12'46″ = 2.07332304365 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 83 ; ; b = 34.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34.8**2 = 83**2 + c**2 -2 * 83 * c * cos (17° ) ; ; ; ; c**2 -158.747c +5677.96 =0 ; ; p=1; q=-158.747; r=5677.96 ; ; D = q**2 - 4pr = 158.747**2 - 4 * 1 * 5677.96 = 2488.63967466 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 158.75 ± sqrt{ 2488.64 } }{ 2 } ; ; c_{1,2} = 79.37329474 ± 24.9431336978 ; ; c_{1} = 104.316428438 ; ;
c_{2} = 54.4301610422 ; ; ; ; text{ Factored form: } ; ; (c -104.316428438) (c -54.4301610422) = 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 = 83 ; ; b = 34.8 ; ; c = 104.32 ; ;

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

p = a+b+c = 83+34.8+104.32 = 222.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 222.12 }{ 2 } = 111.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 111.06 * (111.06-83)(111.06-34.8)(111.06-104.32) } ; ; T = sqrt{ 1602036.08 } = 1265.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1265.72 }{ 83 } = 30.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1265.72 }{ 34.8 } = 72.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1265.72 }{ 104.32 } = 24.27 ; ;

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{ 34.8**2+104.32**2-83**2 }{ 2 * 34.8 * 104.32 } ) = 44° 12'46" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 83**2+104.32**2-34.8**2 }{ 2 * 83 * 104.32 } ) = 17° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 83**2+34.8**2-104.32**2 }{ 2 * 83 * 34.8 } ) = 118° 47'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1265.72 }{ 111.06 } = 11.4 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 83 }{ 2 * sin 44° 12'46" } = 59.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.8**2+2 * 104.32**2 - 83**2 } }{ 2 } = 65.759 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 104.32**2+2 * 83**2 - 34.8**2 } }{ 2 } = 92.643 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.8**2+2 * 83**2 - 104.32**2 } }{ 2 } = 36.463 ; ;







#2 Obtuse scalene triangle.

Sides: a = 83   b = 34.8   c = 54.43301610471

Area: T = 660.4244317408
Perimeter: p = 172.2330161047
Semiperimeter: s = 86.11550805236

Angle ∠ A = α = 135.7877353967° = 135°47'14″ = 2.37699364093 rad
Angle ∠ B = β = 17° = 0.29767059728 rad
Angle ∠ C = γ = 27.21326460331° = 27°12'46″ = 0.47549502715 rad

Height: ha = 15.91438389737
Height: hb = 37.95554205407
Height: hc = 24.2676851492

Median: ma = 19.09442718062
Median: mb = 67.99330968247
Median: mc = 57.5277031838

Inradius: r = 7.66990901686
Circumradius: R = 59.51332829851

Vertex coordinates: A[54.43301610471; 0] B[0; 0] C[79.37332947449; 24.2676851492]
Centroid: CG[44.60111519307; 8.08989504973]
Coordinates of the circumscribed circle: U[27.21550805236; 52.92660828303]
Coordinates of the inscribed circle: I[51.31550805236; 7.66990901686]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 44.21326460331° = 44°12'46″ = 2.37699364093 rad
∠ B' = β' = 163° = 0.29767059728 rad
∠ C' = γ' = 152.7877353967° = 152°47'14″ = 0.47549502715 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 83 ; ; b = 34.8 ; ; beta = 17° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34.8**2 = 83**2 + c**2 -2 * 83 * c * cos (17° ) ; ; ; ; c**2 -158.747c +5677.96 =0 ; ; p=1; q=-158.747; r=5677.96 ; ; D = q**2 - 4pr = 158.747**2 - 4 * 1 * 5677.96 = 2488.63967466 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 158.75 ± sqrt{ 2488.64 } }{ 2 } ; ; c_{1,2} = 79.37329474 ± 24.9431336978 ; ; c_{1} = 104.316428438 ; ; : Nr. 1
c_{2} = 54.4301610422 ; ; ; ; text{ Factored form: } ; ; (c -104.316428438) (c -54.4301610422) = 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 = 83 ; ; b = 34.8 ; ; c = 54.43 ; ;

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

p = a+b+c = 83+34.8+54.43 = 172.23 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 172.23 }{ 2 } = 86.12 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 86.12 * (86.12-83)(86.12-34.8)(86.12-54.43) } ; ; T = sqrt{ 436160.28 } = 660.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 660.42 }{ 83 } = 15.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 660.42 }{ 34.8 } = 37.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 660.42 }{ 54.43 } = 24.27 ; ;

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{ 34.8**2+54.43**2-83**2 }{ 2 * 34.8 * 54.43 } ) = 135° 47'14" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 83**2+54.43**2-34.8**2 }{ 2 * 83 * 54.43 } ) = 17° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 83**2+34.8**2-54.43**2 }{ 2 * 83 * 34.8 } ) = 27° 12'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 660.42 }{ 86.12 } = 7.67 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 83 }{ 2 * sin 135° 47'14" } = 59.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.8**2+2 * 54.43**2 - 83**2 } }{ 2 } = 19.094 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 54.43**2+2 * 83**2 - 34.8**2 } }{ 2 } = 67.993 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.8**2+2 * 83**2 - 54.43**2 } }{ 2 } = 57.527 ; ;
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