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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

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




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