Triangle calculator SSA

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Triangle has two solutions with side c=57.91441161279 and with side c=32.47991281602

#1 Acute scalene triangle.

Sides: a = 59   b = 40   c = 57.91441161279

Area: T = 1098.181105005
Perimeter: p = 156.9144116128
Semiperimeter: s = 78.45770580639

Angle ∠ A = α = 71.46217696984° = 71°27'42″ = 1.24772431705 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 68.53882303016° = 68°32'18″ = 1.19662177823 rad

Height: ha = 37.22664762729
Height: hb = 54.90990525026
Height: hc = 37.92444689715

Median: ma = 40.08545659006
Median: mb = 54.93219799701
Median: mc = 41.25551668071

Inradius: r = 13.99772244327
Circumradius: R = 31.11444765372

Vertex coordinates: A[57.91441161279; 0] B[0; 0] C[45.1976622144; 37.92444689715]
Centroid: CG[34.37702460906; 12.64114896572]
Coordinates of the circumscribed circle: U[28.95770580639; 11.38441749137]
Coordinates of the inscribed circle: I[38.45770580639; 13.99772244327]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.5388230302° = 108°32'18″ = 1.24772431705 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 111.4621769698° = 111°27'42″ = 1.19662177823 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 59 ; ; b = 40 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 59**2 + c**2 -2 * 59 * c * cos (40° ) ; ; ; ; c**2 -90.393c +1881 =0 ; ; p=1; q=-90.393; r=1881 ; ; D = q**2 - 4pr = 90.393**2 - 4 * 1 * 1881 = 646.938612917 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 90.39 ± sqrt{ 646.94 } }{ 2 } ; ; c_{1,2} = 45.19662214 ± 12.7174939839 ; ; c_{1} = 57.9141161239 ; ;
c_{2} = 32.4791281561 ; ; ; ; (c -57.9141161239) (c -32.4791281561) = 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 = 59 ; ; b = 40 ; ; c = 57.91 ; ;

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

p = a+b+c = 59+40+57.91 = 156.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156.91 }{ 2 } = 78.46 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.46 * (78.46-59)(78.46-40)(78.46-57.91) } ; ; T = sqrt{ 1206001.62 } = 1098.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1098.18 }{ 59 } = 37.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1098.18 }{ 40 } = 54.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1098.18 }{ 57.91 } = 37.92 ; ;

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{ 59**2-40**2-57.91**2 }{ 2 * 40 * 57.91 } ) = 71° 27'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-59**2-57.91**2 }{ 2 * 59 * 57.91 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 57.91**2-59**2-40**2 }{ 2 * 40 * 59 } ) = 68° 32'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1098.18 }{ 78.46 } = 14 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 59 }{ 2 * sin 71° 27'42" } = 31.11 ; ;





#2 Obtuse scalene triangle.

Sides: a = 59   b = 40   c = 32.47991281602

Area: T = 615.8776844066
Perimeter: p = 131.479912816
Semiperimeter: s = 65.74395640801

Angle ∠ A = α = 108.5388230302° = 108°32'18″ = 1.89443494831 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 31.46217696984° = 31°27'42″ = 0.54991114697 rad

Height: ha = 20.87771811548
Height: hb = 30.79438422033
Height: hc = 37.92444689715

Median: ma = 21.38221627302
Median: mb = 43.22197510754
Median: mc = 47.71655798297

Inradius: r = 9.36884351681
Circumradius: R = 31.11444765372

Vertex coordinates: A[32.47991281602; 0] B[0; 0] C[45.1976622144; 37.92444689715]
Centroid: CG[25.89219167681; 12.64114896572]
Coordinates of the circumscribed circle: U[16.24395640801; 26.54402940578]
Coordinates of the inscribed circle: I[25.74395640801; 9.36884351681]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 71.46217696984° = 71°27'42″ = 1.89443494831 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 148.5388230302° = 148°32'18″ = 0.54991114697 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 59 ; ; b = 40 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 59**2 + c**2 -2 * 59 * c * cos (40° ) ; ; ; ; c**2 -90.393c +1881 =0 ; ; p=1; q=-90.393; r=1881 ; ; D = q**2 - 4pr = 90.393**2 - 4 * 1 * 1881 = 646.938612917 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 90.39 ± sqrt{ 646.94 } }{ 2 } ; ; c_{1,2} = 45.19662214 ± 12.7174939839 ; ; c_{1} = 57.9141161239 ; ; : Nr. 1
c_{2} = 32.4791281561 ; ; ; ; (c -57.9141161239) (c -32.4791281561) = 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 = 59 ; ; b = 40 ; ; c = 32.48 ; ;

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

p = a+b+c = 59+40+32.48 = 131.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.48 }{ 2 } = 65.74 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.74 * (65.74-59)(65.74-40)(65.74-32.48) } ; ; T = sqrt{ 379304.29 } = 615.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 615.88 }{ 59 } = 20.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 615.88 }{ 40 } = 30.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 615.88 }{ 32.48 } = 37.92 ; ;

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{ 59**2-40**2-32.48**2 }{ 2 * 40 * 32.48 } ) = 108° 32'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-59**2-32.48**2 }{ 2 * 59 * 32.48 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 32.48**2-59**2-40**2 }{ 2 * 40 * 59 } ) = 31° 27'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 615.88 }{ 65.74 } = 9.37 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 59 }{ 2 * sin 108° 32'18" } = 31.11 ; ;




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