Triangle calculator SSA

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Triangle has two solutions with side c=45.39663754419 and with side c=38.85877277994

#1 Acute scalene triangle.

Sides: a = 70   b = 56   c = 45.39663754419

Area: T = 1268.931051094
Perimeter: p = 171.3966375442
Semiperimeter: s = 85.6988187721

Angle ∠ A = α = 86.65331245162° = 86°39'11″ = 1.51223823299 rad
Angle ∠ B = β = 53° = 0.92550245036 rad
Angle ∠ C = γ = 40.34768754838° = 40°20'49″ = 0.70441858201 rad

Height: ha = 36.25551574554
Height: hb = 45.31989468192
Height: hc = 55.90444857033

Median: ma = 37.06596202305
Median: mb = 51.92770204386
Median: mc = 59.18443921502

Inradius: r = 14.80769701902
Circumradius: R = 35.06597984284

Vertex coordinates: A[45.39663754419; 0] B[0; 0] C[42.12770516206; 55.90444857033]
Centroid: CG[29.17444756875; 18.63548285678]
Coordinates of the circumscribed circle: U[22.6988187721; 26.72204367483]
Coordinates of the inscribed circle: I[29.6988187721; 14.80769701902]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 93.34768754838° = 93°20'49″ = 1.51223823299 rad
∠ B' = β' = 127° = 0.92550245036 rad
∠ C' = γ' = 139.6533124516° = 139°39'11″ = 0.70441858201 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 70 ; ; b = 56 ; ; beta = 53° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 70**2 + c**2 -2 * 70 * c * cos (53° ) ; ; ; ; c**2 -84.254c +1764 =0 ; ; p=1; q=-84.254; r=1764 ; ; D = q**2 - 4pr = 84.254**2 - 4 * 1 * 1764 = 42.7539129934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 84.25 ± sqrt{ 42.75 } }{ 2 } ; ; c_{1,2} = 42.12705162 ± 3.26932382127 ; ; c_{1} = 45.3963754413 ; ;
c_{2} = 38.8577277987 ; ; ; ; text{ Factored form: } ; ; (c -45.3963754413) (c -38.8577277987) = 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 = 70 ; ; b = 56 ; ; c = 45.4 ; ;

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

p = a+b+c = 70+56+45.4 = 171.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 171.4 }{ 2 } = 85.7 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 85.7 * (85.7-70)(85.7-56)(85.7-45.4) } ; ; T = sqrt{ 1610184.64 } = 1268.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1268.93 }{ 70 } = 36.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1268.93 }{ 56 } = 45.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1268.93 }{ 45.4 } = 55.9 ; ;

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{ 56**2+45.4**2-70**2 }{ 2 * 56 * 45.4 } ) = 86° 39'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 70**2+45.4**2-56**2 }{ 2 * 70 * 45.4 } ) = 53° ; ; gamma = 180° - alpha - beta = 180° - 86° 39'11" - 53° = 40° 20'49" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1268.93 }{ 85.7 } = 14.81 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 70 }{ 2 * sin 86° 39'11" } = 35.06 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 45.4**2 - 70**2 } }{ 2 } = 37.06 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.4**2+2 * 70**2 - 56**2 } }{ 2 } = 51.927 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 70**2 - 45.4**2 } }{ 2 } = 59.184 ; ;







#2 Obtuse scalene triangle.

Sides: a = 70   b = 56   c = 38.85877277994

Area: T = 1086.161064411
Perimeter: p = 164.8587727799
Semiperimeter: s = 82.42988638997

Angle ∠ A = α = 93.34768754838° = 93°20'49″ = 1.62992103236 rad
Angle ∠ B = β = 53° = 0.92550245036 rad
Angle ∠ C = γ = 33.65331245162° = 33°39'11″ = 0.58773578264 rad

Height: ha = 31.03331612603
Height: hb = 38.79114515754
Height: hc = 55.90444857033

Median: ma = 33.13655021822
Median: mb = 49.20332672174
Median: mc = 60.33767155849

Inradius: r = 13.17769454621
Circumradius: R = 35.06597984284

Vertex coordinates: A[38.85877277994; 0] B[0; 0] C[42.12770516206; 55.90444857033]
Centroid: CG[26.99549264733; 18.63548285678]
Coordinates of the circumscribed circle: U[19.42988638997; 29.1844048955]
Coordinates of the inscribed circle: I[26.42988638997; 13.17769454621]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 86.65331245162° = 86°39'11″ = 1.62992103236 rad
∠ B' = β' = 127° = 0.92550245036 rad
∠ C' = γ' = 146.3476875484° = 146°20'49″ = 0.58773578264 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 70 ; ; b = 56 ; ; beta = 53° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 70**2 + c**2 -2 * 70 * c * cos (53° ) ; ; ; ; c**2 -84.254c +1764 =0 ; ; p=1; q=-84.254; r=1764 ; ; D = q**2 - 4pr = 84.254**2 - 4 * 1 * 1764 = 42.7539129934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 84.25 ± sqrt{ 42.75 } }{ 2 } ; ; c_{1,2} = 42.12705162 ± 3.26932382127 ; ; c_{1} = 45.3963754413 ; ; : Nr. 1
c_{2} = 38.8577277987 ; ; ; ; text{ Factored form: } ; ; (c -45.3963754413) (c -38.8577277987) = 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 = 70 ; ; b = 56 ; ; c = 38.86 ; ;

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

p = a+b+c = 70+56+38.86 = 164.86 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 164.86 }{ 2 } = 82.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 82.43 * (82.43-70)(82.43-56)(82.43-38.86) } ; ; T = sqrt{ 1179744.94 } = 1086.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1086.16 }{ 70 } = 31.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1086.16 }{ 56 } = 38.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1086.16 }{ 38.86 } = 55.9 ; ;

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{ 56**2+38.86**2-70**2 }{ 2 * 56 * 38.86 } ) = 93° 20'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 70**2+38.86**2-56**2 }{ 2 * 70 * 38.86 } ) = 53° ; ; gamma = 180° - alpha - beta = 180° - 93° 20'49" - 53° = 33° 39'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1086.16 }{ 82.43 } = 13.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 70 }{ 2 * sin 93° 20'49" } = 35.06 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 38.86**2 - 70**2 } }{ 2 } = 33.136 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38.86**2+2 * 70**2 - 56**2 } }{ 2 } = 49.203 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 70**2 - 38.86**2 } }{ 2 } = 60.337 ; ;
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