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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 34.8**2 = 83**2 + c**2 -2 * 34.8 * c * cos (17° ) ; ; ; ; c**2 -158.747c +5677.96 =0 ; ; p=1; q=-158.74658949; 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} = 79.3732947449 ± 24.9431336978 ; ;
c_{1} = 104.316428443 ; ; c_{2} = 54.4301610471 ; ; ; ; (c -104.316428443) (c -54.4301610471) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 83**2-34.8**2-54.43**2 }{ 2 * 34.8 * 54.43 } ) = 135° 47'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34.8**2-83**2-54.43**2 }{ 2 * 83 * 54.43 } ) = 17° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54.43**2-83**2-34.8**2 }{ 2 * 34.8 * 83 } ) = 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 ; ;




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