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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 32**2 = 35**2 + c**2 -2 * 32 * c * cos (65° ) ; ; ; ; c**2 -29.583c +201 =0 ; ; p=1; q=-29.5832783218; 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} = 14.7916391609 ± 4.21812625072 ; ; c_{1} = 19.0097654116 ; ;
c_{2} = 10.5735129102 ; ; ; ; (c -19.0097654116) (c -10.5735129102) = 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 = 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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