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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 40**2 = 59**2 + c**2 -2 * 40 * c * cos (40° ) ; ; ; ; c**2 -90.393c +1881 =0 ; ; p=1; q=-90.393244288; 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} = 45.196622144 ± 12.7174939839 ; ; c_{1} = 57.9141161279 ; ;
c_{2} = 32.4791281602 ; ; ; ; (c -57.9141161279) (c -32.4791281602) = 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 = 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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