Triangle calculator SSA

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Triangle has two solutions with side c=83.98662987983 and with side c=11.01437012017

#1 Acute scalene triangle.

Sides: a = 95   b = 90   c = 83.98662987983

Area: T = 3454.878774564
Perimeter: p = 268.9866298798
Semiperimeter: s = 134.4933149399

Angle ∠ A = α = 66.08435962872° = 66°5'1″ = 1.15333763368 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 53.91664037128° = 53°54'59″ = 0.94110187656 rad

Height: ha = 72.73442683292
Height: hb = 76.77550610141
Height: hc = 82.27224133595

Median: ma = 72.94224375307
Median: mb = 77.55222352542
Median: mc = 82.45765061323

Inradius: r = 25.68881317827
Circumradius: R = 51.96215242271

Vertex coordinates: A[83.98662987983; 0] B[0; 0] C[47.5; 82.27224133595]
Centroid: CG[43.82987662661; 27.42441377865]
Coordinates of the circumscribed circle: U[41.99331493992; 30.60435194633]
Coordinates of the inscribed circle: I[44.49331493992; 25.68881317827]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113.9166403713° = 113°54'59″ = 1.15333763368 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 126.0843596287° = 126°5'1″ = 0.94110187656 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 95 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 95**2 + c**2 -2 * 95 * c * cos (60° ) ; ; ; ; c**2 -95c +925 =0 ; ; p=1; q=-95; r=925 ; ; D = q**2 - 4pr = 95**2 - 4 * 1 * 925 = 5325 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 95 ± sqrt{ 5325 } }{ 2 } ; ; c_{1,2} = 47.5 ± 36.4862987983 ; ; c_{1} = 83.9862987983 ; ; c_{2} = 11.0137012017 ; ;
 ; ; text{ Factored form: } ; ; (c -83.9862987983) (c -11.0137012017) = 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 = 95 ; ; b = 90 ; ; c = 83.99 ; ;

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

p = a+b+c = 95+90+83.99 = 268.99 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 268.99 }{ 2 } = 134.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.49 * (134.49-95)(134.49-90)(134.49-83.99) } ; ; T = sqrt{ 11936180.24 } = 3454.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3454.88 }{ 95 } = 72.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3454.88 }{ 90 } = 76.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3454.88 }{ 83.99 } = 82.27 ; ;

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{ 90**2+83.99**2-95**2 }{ 2 * 90 * 83.99 } ) = 66° 5'1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 95**2+83.99**2-90**2 }{ 2 * 95 * 83.99 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 66° 5'1" - 60° = 53° 54'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3454.88 }{ 134.49 } = 25.69 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 95 }{ 2 * sin 66° 5'1" } = 51.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 83.99**2 - 95**2 } }{ 2 } = 72.942 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.99**2+2 * 95**2 - 90**2 } }{ 2 } = 77.552 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 95**2 - 83.99**2 } }{ 2 } = 82.457 ; ;







#2 Obtuse scalene triangle.

Sides: a = 95   b = 90   c = 11.01437012017

Area: T = 453.0621888942
Perimeter: p = 196.0143701202
Semiperimeter: s = 98.00768506008

Angle ∠ A = α = 113.9166403713° = 113°54'59″ = 1.98882163168 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 6.08435962872° = 6°5'1″ = 0.10661787856 rad

Height: ha = 9.53881450303
Height: hb = 10.06880419765
Height: hc = 82.27224133595

Median: ma = 43.06327542905
Median: mb = 50.47992116329
Median: mc = 92.37697710101

Inradius: r = 4.62327573498
Circumradius: R = 51.96215242271

Vertex coordinates: A[11.01437012017; 0] B[0; 0] C[47.5; 82.27224133595]
Centroid: CG[19.50545670672; 27.42441377865]
Coordinates of the circumscribed circle: U[5.50768506008; 51.66988938962]
Coordinates of the inscribed circle: I[8.00768506008; 4.62327573498]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 66.08435962872° = 66°5'1″ = 1.98882163168 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 173.9166403713° = 173°54'59″ = 0.10661787856 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 95 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 95**2 + c**2 -2 * 95 * c * cos (60° ) ; ; ; ; c**2 -95c +925 =0 ; ; p=1; q=-95; r=925 ; ; D = q**2 - 4pr = 95**2 - 4 * 1 * 925 = 5325 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 95 ± sqrt{ 5325 } }{ 2 } ; ; c_{1,2} = 47.5 ± 36.4862987983 ; ; c_{1} = 83.9862987983 ; ; c_{2} = 11.0137012017 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -83.9862987983) (c -11.0137012017) = 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 = 95 ; ; b = 90 ; ; c = 11.01 ; ;

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

p = a+b+c = 95+90+11.01 = 196.01 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 196.01 }{ 2 } = 98.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 98.01 * (98.01-95)(98.01-90)(98.01-11.01) } ; ; T = sqrt{ 205265.08 } = 453.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 453.06 }{ 95 } = 9.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 453.06 }{ 90 } = 10.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 453.06 }{ 11.01 } = 82.27 ; ;

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{ 90**2+11.01**2-95**2 }{ 2 * 90 * 11.01 } ) = 113° 54'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 95**2+11.01**2-90**2 }{ 2 * 95 * 11.01 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 113° 54'59" - 60° = 6° 5'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 453.06 }{ 98.01 } = 4.62 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 95 }{ 2 * sin 113° 54'59" } = 51.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 11.01**2 - 95**2 } }{ 2 } = 43.063 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.01**2+2 * 95**2 - 90**2 } }{ 2 } = 50.479 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 95**2 - 11.01**2 } }{ 2 } = 92.37 ; ;
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