Triangle calculator SSA

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Triangle has two solutions with side c=19.01097654116 and with side c=10.57435129102

#1 Acute scalene triangle.

Sides: a = 35   b = 32   c = 19.01097654116

Area: T = 301.502222239
Perimeter: p = 86.01097654116
Semiperimeter: s = 43.00548827058

Angle ∠ A = α = 82.42554296273° = 82°25'32″ = 1.43985951344 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 32.57545703727° = 32°34'28″ = 0.56985335054 rad

Height: ha = 17.22986984223
Height: hb = 18.84438888994
Height: hc = 31.72107725463

Median: ma = 19.65879650652
Median: mb = 23.17772645173
Median: mc = 32.15883147063

Inradius: r = 7.01108834956
Circumradius: R = 17.65440467034

Vertex coordinates: A[19.01097654116; 0] B[0; 0] C[14.79216391609; 31.72107725463]
Centroid: CG[11.26771348575; 10.57435908488]
Coordinates of the circumscribed circle: U[9.50548827058; 14.87769139863]
Coordinates of the inscribed circle: I[11.00548827058; 7.01108834956]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.57545703727° = 97°34'28″ = 1.43985951344 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 147.4255429627° = 147°25'32″ = 0.56985335054 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 32 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 35**2 + c**2 -2 * 35 * c * cos (65° ) ; ; ; ; c**2 -29.583c +201 =0 ; ; p=1; q=-29.583; r=201 ; ; D = q**2 - 4pr = 29.583**2 - 4 * 1 * 201 = 71.170356268 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 29.58 ± sqrt{ 71.17 } }{ 2 } ; ; c_{1,2} = 14.79163916 ± 4.21812625072 ; ; c_{1} = 19.0097654107 ; ;
c_{2} = 10.5735129093 ; ; ; ; (c -19.0097654107) (c -10.5735129093) = 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 = 35 ; ; b = 32 ; ; c = 19.01 ; ;

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

p = a+b+c = 35+32+19.01 = 86.01 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 86.01 }{ 2 } = 43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43 * (43-35)(43-32)(43-19.01) } ; ; T = sqrt{ 90903.59 } = 301.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 301.5 }{ 35 } = 17.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 301.5 }{ 32 } = 18.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 301.5 }{ 19.01 } = 31.72 ; ;

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{ 35**2-32**2-19.01**2 }{ 2 * 32 * 19.01 } ) = 82° 25'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-35**2-19.01**2 }{ 2 * 35 * 19.01 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19.01**2-35**2-32**2 }{ 2 * 32 * 35 } ) = 32° 34'28" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 301.5 }{ 43 } = 7.01 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 82° 25'32" } = 17.65 ; ;





#2 Obtuse scalene triangle.

Sides: a = 35   b = 32   c = 10.57435129102

Area: T = 167.769999902
Perimeter: p = 77.57435129102
Semiperimeter: s = 38.78767564551

Angle ∠ A = α = 97.57545703727° = 97°34'28″ = 1.70329975192 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 17.42554296273° = 17°25'32″ = 0.30441311206 rad

Height: ha = 9.58328570869
Height: hb = 10.48112499387
Height: hc = 31.72107725463

Median: ma = 16.17655861604
Median: mb = 20.30876238795
Median: mc = 33.11441994646

Inradius: r = 4.32436407049
Circumradius: R = 17.65440467034

Vertex coordinates: A[10.57435129102; 0] B[0; 0] C[14.79216391609; 31.72107725463]
Centroid: CG[8.45550506904; 10.57435908488]
Coordinates of the circumscribed circle: U[5.28767564551; 16.844385856]
Coordinates of the inscribed circle: I[6.78767564551; 4.32436407049]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 82.42554296273° = 82°25'32″ = 1.70329975192 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 162.5754570373° = 162°34'28″ = 0.30441311206 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 32 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 35**2 + c**2 -2 * 35 * c * cos (65° ) ; ; ; ; c**2 -29.583c +201 =0 ; ; p=1; q=-29.583; r=201 ; ; D = q**2 - 4pr = 29.583**2 - 4 * 1 * 201 = 71.170356268 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 29.58 ± sqrt{ 71.17 } }{ 2 } ; ; c_{1,2} = 14.79163916 ± 4.21812625072 ; ; c_{1} = 19.0097654107 ; ; : Nr. 1
c_{2} = 10.5735129093 ; ; ; ; (c -19.0097654107) (c -10.5735129093) = 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 = 35 ; ; b = 32 ; ; c = 10.57 ; ;

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

p = a+b+c = 35+32+10.57 = 77.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77.57 }{ 2 } = 38.79 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.79 * (38.79-35)(38.79-32)(38.79-10.57) } ; ; T = sqrt{ 28123.29 } = 167.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.7 }{ 35 } = 9.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.7 }{ 32 } = 10.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.7 }{ 10.57 } = 31.72 ; ;

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{ 35**2-32**2-10.57**2 }{ 2 * 32 * 10.57 } ) = 97° 34'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-35**2-10.57**2 }{ 2 * 35 * 10.57 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.57**2-35**2-32**2 }{ 2 * 32 * 35 } ) = 17° 25'32" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.7 }{ 38.79 } = 4.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 97° 34'28" } = 17.65 ; ;




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