Triangle calculator SSA

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Triangle has two solutions with side c=19.34438762462 and with side c=12.87222907876

#1 Acute scalene triangle.

Sides: a = 43   b = 40   c = 19.34438762462

Area: T = 385.6109589414
Perimeter: p = 102.3443876246
Semiperimeter: s = 51.17219381231

Angle ∠ A = α = 85.36599982427° = 85°21'36″ = 1.49898130188 rad
Angle ∠ B = β = 68° = 1.18768238914 rad
Angle ∠ C = γ = 26.64400017573° = 26°38'24″ = 0.46549557434 rad

Height: ha = 17.93553297402
Height: hb = 19.28804794707
Height: hc = 39.86989057464

Median: ma = 22.90994472677
Median: mb = 26.67656963192
Median: mc = 40.38550667072

Inradius: r = 7.53655674137
Circumradius: R = 21.57106948536

Vertex coordinates: A[19.34438762462; 0] B[0; 0] C[16.10880835169; 39.86989057464]
Centroid: CG[11.8177319921; 13.29896352488]
Coordinates of the circumscribed circle: U[9.67219381231; 19.28107803112]
Coordinates of the inscribed circle: I[11.17219381231; 7.53655674137]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 94.64400017573° = 94°38'24″ = 1.49898130188 rad
∠ B' = β' = 112° = 1.18768238914 rad
∠ C' = γ' = 153.3659998243° = 153°21'36″ = 0.46549557434 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 = 43 ; ; b = 40 ; ; c = 19.34 ; ;

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

p = a+b+c = 43+40+19.34 = 102.34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 102.34 }{ 2 } = 51.17 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 51.17 * (51.17-43)(51.17-40)(51.17-19.34) } ; ; T = sqrt{ 148694.76 } = 385.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 385.61 }{ 43 } = 17.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 385.61 }{ 40 } = 19.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 385.61 }{ 19.34 } = 39.87 ; ;

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{ 43**2-40**2-19.34**2 }{ 2 * 40 * 19.34 } ) = 85° 21'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-43**2-19.34**2 }{ 2 * 43 * 19.34 } ) = 68° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19.34**2-43**2-40**2 }{ 2 * 40 * 43 } ) = 26° 38'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 385.61 }{ 51.17 } = 7.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 43 }{ 2 * sin 85° 21'36" } = 21.57 ; ;





#2 Obtuse scalene triangle.

Sides: a = 43   b = 40   c = 12.87222907876

Area: T = 256.6022074075
Perimeter: p = 95.87222907876
Semiperimeter: s = 47.93661453938

Angle ∠ A = α = 94.64400017573° = 94°38'24″ = 1.65217796348 rad
Angle ∠ B = β = 68° = 1.18768238914 rad
Angle ∠ C = γ = 17.36599982427° = 17°21'36″ = 0.30329891275 rad

Height: ha = 11.93549801895
Height: hb = 12.83301037038
Height: hc = 39.86989057464

Median: ma = 20.50884844652
Median: mb = 24.64444301022
Median: mc = 41.02553096572

Inradius: r = 5.3532997659
Circumradius: R = 21.57106948536

Vertex coordinates: A[12.87222907876; 0] B[0; 0] C[16.10880835169; 39.86989057464]
Centroid: CG[9.66601247682; 13.29896352488]
Coordinates of the circumscribed circle: U[6.43661453938; 20.58881254352]
Coordinates of the inscribed circle: I[7.93661453938; 5.3532997659]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 85.36599982427° = 85°21'36″ = 1.65217796348 rad
∠ B' = β' = 112° = 1.18768238914 rad
∠ C' = γ' = 162.6440001757° = 162°38'24″ = 0.30329891275 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 43 ; ; b = 40 ; ; beta = 68° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 40**2 = 43**2 + c**2 -2 * 40 * c * cos (68° ) ; ; ; ; c**2 -32.216c +249 =0 ; ; p=1; q=-32.2161670338; r=249 ; ; D = q**2 - 4pr = 32.216**2 - 4 * 1 * 249 = 41.8814183477 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 32.22 ± sqrt{ 41.88 } }{ 2 } ; ; c_{1,2} = 16.1080835169 ± 3.23579272929 ; ; c_{1} = 19.3438762462 ; ;
c_{2} = 12.8722907876 ; ; ; ; (c -19.3438762462) (c -12.8722907876) = 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 = 43 ; ; b = 40 ; ; c = 12.87 ; ;

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

p = a+b+c = 43+40+12.87 = 95.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 95.87 }{ 2 } = 47.94 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 47.94 * (47.94-43)(47.94-40)(47.94-12.87) } ; ; T = sqrt{ 65844.62 } = 256.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 256.6 }{ 43 } = 11.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 256.6 }{ 40 } = 12.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 256.6 }{ 12.87 } = 39.87 ; ;

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{ 43**2-40**2-12.87**2 }{ 2 * 40 * 12.87 } ) = 94° 38'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-43**2-12.87**2 }{ 2 * 43 * 12.87 } ) = 68° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.87**2-43**2-40**2 }{ 2 * 40 * 43 } ) = 17° 21'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 256.6 }{ 47.94 } = 5.35 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 43 }{ 2 * sin 94° 38'24" } = 21.57 ; ;




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