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?

1. Use Law of Cosines

a = 43 ; ; b = 40 ; ; beta = 68° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 43**2 + c**2 -2 * 43 * c * cos (68° ) ; ; ; ; c**2 -32.216c +249 =0 ; ; p=1; q=-32.216; 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} = 16.10808352 ± 3.23579272929 ; ; c_{1} = 19.3438762462 ; ; c_{2} = 12.8722907876 ; ; ; ; text{ Factored form: } ; ; (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 = 19.34 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 40**2+19.34**2-43**2 }{ 2 * 40 * 19.34 } ) = 85° 21'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43**2+19.34**2-40**2 }{ 2 * 43 * 19.34 } ) = 68° ; ;
 gamma = 180° - alpha - beta = 180° - 85° 21'36" - 68° = 26° 38'24" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 19.34**2 - 43**2 } }{ 2 } = 22.909 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.34**2+2 * 43**2 - 40**2 } }{ 2 } = 26.676 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 43**2 - 19.34**2 } }{ 2 } = 40.385 ; ;



#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 - 2ac cos beta ; ; 40**2 = 43**2 + c**2 -2 * 43 * c * cos (68° ) ; ; ; ; c**2 -32.216c +249 =0 ; ; p=1; q=-32.216; 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 } ; ; : Nr. 1
c_{1,2} = 16.10808352 ± 3.23579272929 ; ; c_{1} = 19.3438762462 ; ; c_{2} = 12.8722907876 ; ; ; ; text{ Factored form: } ; ; (c -19.3438762462) (c -12.8722907876) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 40**2+12.87**2-43**2 }{ 2 * 40 * 12.87 } ) = 94° 38'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43**2+12.87**2-40**2 }{ 2 * 43 * 12.87 } ) = 68° ; ;
 gamma = 180° - alpha - beta = 180° - 94° 38'24" - 68° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 12.87**2 - 43**2 } }{ 2 } = 20.508 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.87**2+2 * 43**2 - 40**2 } }{ 2 } = 24.644 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 43**2 - 12.87**2 } }{ 2 } = 41.025 ; ;
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