Triangle calculator SSA

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Triangle has two solutions with side c=80.92987410057 and with side c=12.35765495716

#1 Obtuse scalene triangle.

Sides: a = 55   b = 45   c = 80.92987410057

Area: T = 1179.357671945
Perimeter: p = 180.9298741006
Semiperimeter: s = 90.46443705028

Angle ∠ A = α = 40.36768437134° = 40°22'1″ = 0.70545343314 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 107.6333156287° = 107°37'59″ = 1.87985529615 rad

Height: ha = 42.8865698889
Height: hb = 52.41658541977
Height: hc = 29.14655595328

Median: ma = 59.42220544948
Median: mb = 65.42992026574
Median: mc = 29.79331992208

Inradius: r = 13.03766984581
Circumradius: R = 42.4599298083

Vertex coordinates: A[80.92987410057; 0] B[0; 0] C[46.64326452886; 29.14655595328]
Centroid: CG[42.52437954314; 9.71551865109]
Coordinates of the circumscribed circle: U[40.46443705028; -12.86218316546]
Coordinates of the inscribed circle: I[45.46443705028; 13.03766984581]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.6333156287° = 139°37'59″ = 0.70545343314 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 72.36768437134° = 72°22'1″ = 1.87985529615 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 = 55 ; ; b = 45 ; ; c = 80.93 ; ;

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

p = a+b+c = 55+45+80.93 = 180.93 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180.93 }{ 2 } = 90.46 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90.46 * (90.46-55)(90.46-45)(90.46-80.93) } ; ; T = sqrt{ 1390882.27 } = 1179.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1179.36 }{ 55 } = 42.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1179.36 }{ 45 } = 52.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1179.36 }{ 80.93 } = 29.15 ; ;

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{ 55**2-45**2-80.93**2 }{ 2 * 45 * 80.93 } ) = 40° 22'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-55**2-80.93**2 }{ 2 * 55 * 80.93 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 80.93**2-55**2-45**2 }{ 2 * 45 * 55 } ) = 107° 37'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1179.36 }{ 90.46 } = 13.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 55 }{ 2 * sin 40° 22'1" } = 42.46 ; ;





#2 Obtuse scalene triangle.

Sides: a = 55   b = 45   c = 12.35765495716

Area: T = 180.0699275579
Perimeter: p = 112.3576549572
Semiperimeter: s = 56.17882747858

Angle ∠ A = α = 139.6333156287° = 139°37'59″ = 2.43770583222 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 8.36768437134° = 8°22'1″ = 0.14660289708 rad

Height: ha = 6.54879736574
Height: hb = 8.00330789146
Height: hc = 29.14655595328

Median: ma = 18.23771093833
Median: mb = 32.90327682522
Median: mc = 49.86881152709

Inradius: r = 3.20553187156
Circumradius: R = 42.4599298083

Vertex coordinates: A[12.35765495716; 0] B[0; 0] C[46.64326452886; 29.14655595328]
Centroid: CG[19.66663982867; 9.71551865109]
Coordinates of the circumscribed circle: U[6.17882747858; 42.00773911874]
Coordinates of the inscribed circle: I[11.17882747858; 3.20553187156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 40.36768437134° = 40°22'1″ = 2.43770583222 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 171.6333156287° = 171°37'59″ = 0.14660289708 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 55 ; ; b = 45 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 45**2 = 55**2 + c**2 -2 * 45 * c * cos (32° ) ; ; ; ; c**2 -93.285c +1000 =0 ; ; p=1; q=-93.2852905772; r=1000 ; ; D = q**2 - 4pr = 93.285**2 - 4 * 1 * 1000 = 4702.14543807 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 93.29 ± sqrt{ 4702.15 } }{ 2 } ; ; c_{1,2} = 46.6426452886 ± 34.286095717 ; ; c_{1} = 80.9287410056 ; ;
c_{2} = 12.3565495716 ; ; ; ; (c -80.9287410056) (c -12.3565495716) = 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 = 55 ; ; b = 45 ; ; c = 12.36 ; ;

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

p = a+b+c = 55+45+12.36 = 112.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 112.36 }{ 2 } = 56.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.18 * (56.18-55)(56.18-45)(56.18-12.36) } ; ; T = sqrt{ 32424.94 } = 180.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 180.07 }{ 55 } = 6.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 180.07 }{ 45 } = 8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 180.07 }{ 12.36 } = 29.15 ; ;

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{ 55**2-45**2-12.36**2 }{ 2 * 45 * 12.36 } ) = 139° 37'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 45**2-55**2-12.36**2 }{ 2 * 55 * 12.36 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.36**2-55**2-45**2 }{ 2 * 45 * 55 } ) = 8° 22'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 180.07 }{ 56.18 } = 3.21 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 55 }{ 2 * sin 139° 37'59" } = 42.46 ; ;




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