Triangle calculator SSA

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Triangle has two solutions with side c=27.32550375794 and with side c=3.31112488771

#1 Obtuse scalene triangle.

Sides: a = 18.7   b = 16.1   c = 27.32550375794

Area: T = 146.5432528289
Perimeter: p = 62.12550375794
Semiperimeter: s = 31.06325187897

Angle ∠ A = α = 41.77547353587° = 41°46'29″ = 0.72991066762 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 103.2255264641° = 103°13'31″ = 1.80216207392 rad

Height: ha = 15.67329976779
Height: hb = 18.20440407812
Height: hc = 10.72658793598

Median: ma = 20.38440952548
Median: mb = 21.98657076156
Median: mc = 10.8532906538

Inradius: r = 4.7187664053
Circumradius: R = 14.03547467047

Vertex coordinates: A[27.32550375794; 0] B[0; 0] C[15.31881432282; 10.72658793598]
Centroid: CG[14.21443936025; 3.57552931199]
Coordinates of the circumscribed circle: U[13.66325187897; -3.21108714375]
Coordinates of the inscribed circle: I[14.96325187897; 4.7187664053]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2255264641° = 138°13'31″ = 0.72991066762 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 76.77547353587° = 76°46'29″ = 1.80216207392 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 18.7 ; ; b = 16.1 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.1**2 = 18.7**2 + c**2 -2 * 18.7 * c * cos (35° ) ; ; ; ; c**2 -30.636c +90.48 =0 ; ; p=1; q=-30.636; r=90.48 ; ; D = q**2 - 4pr = 30.636**2 - 4 * 1 * 90.48 = 576.662047839 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.64 ± sqrt{ 576.66 } }{ 2 } ; ; c_{1,2} = 15.31814323 ± 12.0068943512 ; ; c_{1} = 27.3250375812 ; ; c_{2} = 3.31124887885 ; ; ; ; text{ Factored form: } ; ; (c -27.3250375812) (c -3.31124887885) = 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 = 18.7 ; ; b = 16.1 ; ; c = 27.33 ; ;

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

p = a+b+c = 18.7+16.1+27.33 = 62.13 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.13 }{ 2 } = 31.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.06 * (31.06-18.7)(31.06-16.1)(31.06-27.33) } ; ; T = sqrt{ 21474.71 } = 146.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.54 }{ 18.7 } = 15.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.54 }{ 16.1 } = 18.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.54 }{ 27.33 } = 10.73 ; ;

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{ 16.1**2+27.33**2-18.7**2 }{ 2 * 16.1 * 27.33 } ) = 41° 46'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.7**2+27.33**2-16.1**2 }{ 2 * 18.7 * 27.33 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 41° 46'29" - 35° = 103° 13'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.54 }{ 31.06 } = 4.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.7 }{ 2 * sin 41° 46'29" } = 14.03 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 27.33**2 - 18.7**2 } }{ 2 } = 20.384 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.33**2+2 * 18.7**2 - 16.1**2 } }{ 2 } = 21.986 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 18.7**2 - 27.33**2 } }{ 2 } = 10.853 ; ;







#2 Obtuse scalene triangle.

Sides: a = 18.7   b = 16.1   c = 3.31112488771

Area: T = 17.75880279927
Perimeter: p = 38.11112488771
Semiperimeter: s = 19.05656244385

Angle ∠ A = α = 138.2255264641° = 138°13'31″ = 2.41224859774 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 6.77547353587° = 6°46'29″ = 0.1188241438 rad

Height: ha = 1.89992543308
Height: hb = 2.20659662103
Height: hc = 10.72658793598

Median: ma = 6.90439615123
Median: mb = 10.74882409985
Median: mc = 17.37697699386

Inradius: r = 0.93219048058
Circumradius: R = 14.03547467047

Vertex coordinates: A[3.31112488771; 0] B[0; 0] C[15.31881432282; 10.72658793598]
Centroid: CG[6.21097973684; 3.57552931199]
Coordinates of the circumscribed circle: U[1.65656244385; 13.93767507973]
Coordinates of the inscribed circle: I[2.95656244385; 0.93219048058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.77547353587° = 41°46'29″ = 2.41224859774 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 173.2255264641° = 173°13'31″ = 0.1188241438 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 18.7 ; ; b = 16.1 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.1**2 = 18.7**2 + c**2 -2 * 18.7 * c * cos (35° ) ; ; ; ; c**2 -30.636c +90.48 =0 ; ; p=1; q=-30.636; r=90.48 ; ; D = q**2 - 4pr = 30.636**2 - 4 * 1 * 90.48 = 576.662047839 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.64 ± sqrt{ 576.66 } }{ 2 } ; ; c_{1,2} = 15.31814323 ± 12.0068943512 ; ; c_{1} = 27.3250375812 ; ; c_{2} = 3.31124887885 ; ; ; ; text{ Factored form: } ; ; (c -27.3250375812) (c -3.31124887885) = 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 = 18.7 ; ; b = 16.1 ; ; c = 3.31 ; ;

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

p = a+b+c = 18.7+16.1+3.31 = 38.11 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.11 }{ 2 } = 19.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.06 * (19.06-18.7)(19.06-16.1)(19.06-3.31) } ; ; T = sqrt{ 315.35 } = 17.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.76 }{ 18.7 } = 1.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.76 }{ 16.1 } = 2.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.76 }{ 3.31 } = 10.73 ; ;

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{ 16.1**2+3.31**2-18.7**2 }{ 2 * 16.1 * 3.31 } ) = 138° 13'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.7**2+3.31**2-16.1**2 }{ 2 * 18.7 * 3.31 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 138° 13'31" - 35° = 6° 46'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.76 }{ 19.06 } = 0.93 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.7 }{ 2 * sin 138° 13'31" } = 14.03 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 3.31**2 - 18.7**2 } }{ 2 } = 6.904 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.31**2+2 * 18.7**2 - 16.1**2 } }{ 2 } = 10.748 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.1**2+2 * 18.7**2 - 3.31**2 } }{ 2 } = 17.37 ; ;
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