Triangle calculator SSA

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Triangle has two solutions with side c=12.88222418628 and with side c=3.04329486123

#1 Acute scalene triangle.

Sides: a = 10.8   b = 8.8   c = 12.88222418628

Area: T = 46.9976828855
Perimeter: p = 32.48222418628
Semiperimeter: s = 16.24111209314

Angle ∠ A = α = 56.01098236305° = 56°35″ = 0.97875558358 rad
Angle ∠ B = β = 42.5° = 42°30' = 0.74217649321 rad
Angle ∠ C = γ = 81.49901763695° = 81°29'25″ = 1.42222718857 rad

Height: ha = 8.70331164546
Height: hb = 10.6811097467
Height: hc = 7.29663742422

Median: ma = 9.62195674386
Median: mb = 11.04224670117
Median: mc = 7.45333188009

Inradius: r = 2.89436936714
Circumradius: R = 6.51328238249

Vertex coordinates: A[12.88222418628; 0] B[0; 0] C[7.96325952375; 7.29663742422]
Centroid: CG[6.94882790334; 2.43221247474]
Coordinates of the circumscribed circle: U[6.44111209314; 0.96437610289]
Coordinates of the inscribed circle: I[7.44111209314; 2.89436936714]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.9990176369° = 123°59'25″ = 0.97875558358 rad
∠ B' = β' = 137.5° = 137°30' = 0.74217649321 rad
∠ C' = γ' = 98.51098236305° = 98°30'35″ = 1.42222718857 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 10.8 ; ; b = 8.8 ; ; beta = 42° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.8**2 = 10.8**2 + c**2 -2 * 10.8 * c * cos (42° 30') ; ; ; ; c**2 -15.925c +39.2 =0 ; ; p=1; q=-15.925; r=39.2 ; ; D = q**2 - 4pr = 15.925**2 - 4 * 1 * 39.2 = 96.8116916682 ; ; D>0 ; ; ; ;
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.93 ± sqrt{ 96.81 } }{ 2 } ; ; c_{1,2} = 7.96259524 ± 4.91964662522 ; ; c_{1} = 12.8822418628 ; ; c_{2} = 3.04294861233 ; ; ; ; text{ Factored form: } ; ; (c -12.8822418628) (c -3.04294861233) = 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 = 10.8 ; ; b = 8.8 ; ; c = 12.88 ; ;

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

p = a+b+c = 10.8+8.8+12.88 = 32.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.48 }{ 2 } = 16.24 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.24 * (16.24-10.8)(16.24-8.8)(16.24-12.88) } ; ; T = sqrt{ 2208.7 } = 47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 47 }{ 10.8 } = 8.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 47 }{ 8.8 } = 10.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 47 }{ 12.88 } = 7.3 ; ;

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{ 8.8**2+12.88**2-10.8**2 }{ 2 * 8.8 * 12.88 } ) = 56° 35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.8**2+12.88**2-8.8**2 }{ 2 * 10.8 * 12.88 } ) = 42° 30' ; ;
 gamma = 180° - alpha - beta = 180° - 56° 35" - 42° 30' = 81° 29'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 47 }{ 16.24 } = 2.89 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.8 }{ 2 * sin 56° 35" } = 6.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.8**2+2 * 12.88**2 - 10.8**2 } }{ 2 } = 9.62 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.88**2+2 * 10.8**2 - 8.8**2 } }{ 2 } = 11.042 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.8**2+2 * 10.8**2 - 12.88**2 } }{ 2 } = 7.453 ; ;



#2 Obtuse scalene triangle.

Sides: a = 10.8   b = 8.8   c = 3.04329486123

Area: T = 11.10112459377
Perimeter: p = 22.64329486123
Semiperimeter: s = 11.32114743062

Angle ∠ A = α = 123.9990176369° = 123°59'25″ = 2.16440368178 rad
Angle ∠ B = β = 42.5° = 42°30' = 0.74217649321 rad
Angle ∠ C = γ = 13.51098236305° = 13°30'35″ = 0.23657909037 rad

Height: ha = 2.05657862848
Height: hb = 2.52330104404
Height: hc = 7.29663742422

Median: ma = 3.76769308633
Median: mb = 6.60222547761
Median: mc = 9.73326828745

Inradius: r = 0.98105477306
Circumradius: R = 6.51328238249

Vertex coordinates: A[3.04329486123; 0] B[0; 0] C[7.96325952375; 7.29663742422]
Centroid: CG[3.66985146166; 2.43221247474]
Coordinates of the circumscribed circle: U[1.52114743062; 6.33326132133]
Coordinates of the inscribed circle: I[2.52114743062; 0.98105477306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.01098236305° = 56°35″ = 2.16440368178 rad
∠ B' = β' = 137.5° = 137°30' = 0.74217649321 rad
∠ C' = γ' = 166.4990176369° = 166°29'25″ = 0.23657909037 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 10.8 ; ; b = 8.8 ; ; beta = 42° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.8**2 = 10.8**2 + c**2 -2 * 10.8 * c * cos (42° 30') ; ; ; ; c**2 -15.925c +39.2 =0 ; ; p=1; q=-15.925; r=39.2 ; ; D = q**2 - 4pr = 15.925**2 - 4 * 1 * 39.2 = 96.8116916682 ; ; D>0 ; ; ; ; : Nr. 1
c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.93 ± sqrt{ 96.81 } }{ 2 } ; ; c_{1,2} = 7.96259524 ± 4.91964662522 ; ; c_{1} = 12.8822418628 ; ; c_{2} = 3.04294861233 ; ; ; ; text{ Factored form: } ; ; (c -12.8822418628) (c -3.04294861233) = 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 = 10.8 ; ; b = 8.8 ; ; c = 3.04 ; ;

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

p = a+b+c = 10.8+8.8+3.04 = 22.64 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.64 }{ 2 } = 11.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.32 * (11.32-10.8)(11.32-8.8)(11.32-3.04) } ; ; T = sqrt{ 123.24 } = 11.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.1 }{ 10.8 } = 2.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.1 }{ 8.8 } = 2.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.1 }{ 3.04 } = 7.3 ; ;

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{ 8.8**2+3.04**2-10.8**2 }{ 2 * 8.8 * 3.04 } ) = 123° 59'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.8**2+3.04**2-8.8**2 }{ 2 * 10.8 * 3.04 } ) = 42° 30' ; ;
 gamma = 180° - alpha - beta = 180° - 123° 59'25" - 42° 30' = 13° 30'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.1 }{ 11.32 } = 0.98 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.8 }{ 2 * sin 123° 59'25" } = 6.51 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.8**2+2 * 3.04**2 - 10.8**2 } }{ 2 } = 3.767 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.04**2+2 * 10.8**2 - 8.8**2 } }{ 2 } = 6.602 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.8**2+2 * 10.8**2 - 3.04**2 } }{ 2 } = 9.733 ; ;
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