Triangle calculator SSA

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Triangle has two solutions with side c=3.50771247279 and with side c=0.69328752721

#1 Acute scalene triangle.

Sides: a = 4.2   b = 3.9   c = 3.50771247279

Area: T = 6.37882441281
Perimeter: p = 11.60771247279
Semiperimeter: s = 5.8043562364

Angle ∠ A = α = 68.85105906375° = 68°51'2″ = 1.20216694986 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 51.14994093625° = 51°8'58″ = 0.89327256038 rad

Height: ha = 3.03772591086
Height: hb = 3.27108944247
Height: hc = 3.63773066959

Median: ma = 3.05769530465
Median: mb = 3.3421775266
Median: mc = 3.65437677862

Inradius: r = 1.09990222433
Circumradius: R = 2.25216660498

Vertex coordinates: A[3.50771247279; 0] B[0; 0] C[2.1; 3.63773066959]
Centroid: CG[1.8699041576; 1.21224355653]
Coordinates of the circumscribed circle: U[1.7543562364; 1.41224514277]
Coordinates of the inscribed circle: I[1.9043562364; 1.09990222433]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.1499409362° = 111°8'58″ = 1.20216694986 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 128.8510590638° = 128°51'2″ = 0.89327256038 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 4.2 ; ; b = 3.9 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.9**2 = 4.2**2 + c**2 -2 * 4.2 * c * cos (60° ) ; ; ; ; c**2 -4.2c +2.43 =0 ; ; p=1; q=-4.2; r=2.43 ; ; D = q**2 - 4pr = 4.2**2 - 4 * 1 * 2.43 = 7.92 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.2 ± sqrt{ 7.92 } }{ 2 } ; ; c_{1,2} = 2.1 ± 1.40712472795 ; ; c_{1} = 3.50712472795 ; ; c_{2} = 0.692875272053 ; ;
 ; ; text{ Factored form: } ; ; (c -3.50712472795) (c -0.692875272053) = 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 = 4.2 ; ; b = 3.9 ; ; c = 3.51 ; ;

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

p = a+b+c = 4.2+3.9+3.51 = 11.61 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.61 }{ 2 } = 5.8 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.8 * (5.8-4.2)(5.8-3.9)(5.8-3.51) } ; ; T = sqrt{ 40.68 } = 6.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.38 }{ 4.2 } = 3.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.38 }{ 3.9 } = 3.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.38 }{ 3.51 } = 3.64 ; ;

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{ 3.9**2+3.51**2-4.2**2 }{ 2 * 3.9 * 3.51 } ) = 68° 51'2" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.2**2+3.51**2-3.9**2 }{ 2 * 4.2 * 3.51 } ) = 60° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 4.2**2+3.9**2-3.51**2 }{ 2 * 4.2 * 3.9 } ) = 51° 8'58" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.38 }{ 5.8 } = 1.1 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.2 }{ 2 * sin 68° 51'2" } = 2.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 3.51**2 - 4.2**2 } }{ 2 } = 3.057 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.51**2+2 * 4.2**2 - 3.9**2 } }{ 2 } = 3.342 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 4.2**2 - 3.51**2 } }{ 2 } = 3.654 ; ;







#2 Obtuse scalene triangle.

Sides: a = 4.2   b = 3.9   c = 0.69328752721

Area: T = 1.26600999332
Perimeter: p = 8.79328752721
Semiperimeter: s = 4.3966437636

Angle ∠ A = α = 111.1499409362° = 111°8'58″ = 1.9439923155 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 8.85105906375° = 8°51'2″ = 0.15444719474 rad

Height: ha = 0.66000475873
Height: hb = 0.6466205094
Height: hc = 3.63773066959

Median: ma = 1.85333855701
Median: mb = 2.29329321995
Median: mc = 4.03879426648

Inradius: r = 0.28766184028
Circumradius: R = 2.25216660498

Vertex coordinates: A[0.69328752721; 0] B[0; 0] C[2.1; 3.63773066959]
Centroid: CG[0.9310958424; 1.21224355653]
Coordinates of the circumscribed circle: U[0.3466437636; 2.22548552682]
Coordinates of the inscribed circle: I[0.4966437636; 0.28766184028]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 68.85105906375° = 68°51'2″ = 1.9439923155 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 171.1499409362° = 171°8'58″ = 0.15444719474 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 4.2 ; ; b = 3.9 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.9**2 = 4.2**2 + c**2 -2 * 4.2 * c * cos (60° ) ; ; ; ; c**2 -4.2c +2.43 =0 ; ; p=1; q=-4.2; r=2.43 ; ; D = q**2 - 4pr = 4.2**2 - 4 * 1 * 2.43 = 7.92 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.2 ± sqrt{ 7.92 } }{ 2 } ; ; c_{1,2} = 2.1 ± 1.40712472795 ; ; c_{1} = 3.50712472795 ; ; c_{2} = 0.692875272053 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -3.50712472795) (c -0.692875272053) = 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 = 4.2 ; ; b = 3.9 ; ; c = 0.69 ; ;

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

p = a+b+c = 4.2+3.9+0.69 = 8.79 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.79 }{ 2 } = 4.4 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.4 * (4.4-4.2)(4.4-3.9)(4.4-0.69) } ; ; T = sqrt{ 1.59 } = 1.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.26 }{ 4.2 } = 0.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.26 }{ 3.9 } = 0.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.26 }{ 0.69 } = 3.64 ; ;

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{ 3.9**2+0.69**2-4.2**2 }{ 2 * 3.9 * 0.69 } ) = 111° 8'58" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.2**2+0.69**2-3.9**2 }{ 2 * 4.2 * 0.69 } ) = 60° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 4.2**2+3.9**2-0.69**2 }{ 2 * 4.2 * 3.9 } ) = 8° 51'2" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.26 }{ 4.4 } = 0.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.2 }{ 2 * sin 111° 8'58" } = 2.25 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 0.69**2 - 4.2**2 } }{ 2 } = 1.853 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.69**2+2 * 4.2**2 - 3.9**2 } }{ 2 } = 2.293 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 4.2**2 - 0.69**2 } }{ 2 } = 4.038 ; ;
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