Triangle calculator SSA

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Triangle has two solutions with side c=9.93298460877 and with side c=1.64765512009

#1 Obtuse scalene triangle.

Sides: a = 6.2   b = 4.7   c = 9.93298460877

Area: T = 11.03114696034
Perimeter: p = 20.83298460877
Semiperimeter: s = 10.41549230439

Angle ∠ A = α = 28.21223488407° = 28°12'44″ = 0.4922398377 rad
Angle ∠ B = β = 21° = 0.36765191429 rad
Angle ∠ C = γ = 130.7887651159° = 130°47'16″ = 2.28326751337 rad

Height: ha = 3.55985385818
Height: hb = 4.69442423844
Height: hc = 2.22218812872

Median: ma = 7.12329152503
Median: mb = 7.93771545067
Median: mc = 2.3769501882

Inradius: r = 1.05991983788
Circumradius: R = 6.55875060576

Vertex coordinates: A[9.93298460877; 0] B[0; 0] C[5.78881986443; 2.22218812872]
Centroid: CG[5.2399348244; 0.74106270957]
Coordinates of the circumscribed circle: U[4.96549230439; -4.28437395888]
Coordinates of the inscribed circle: I[5.71549230439; 1.05991983788]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.7887651159° = 151°47'16″ = 0.4922398377 rad
∠ B' = β' = 159° = 0.36765191429 rad
∠ C' = γ' = 49.21223488407° = 49°12'44″ = 2.28326751337 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 6.2 ; ; b = 4.7 ; ; beta = 21° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.7**2 = 6.2**2 + c**2 -2 * 6.2 * c * cos (21° ) ; ; ; ; c**2 -11.576c +16.35 =0 ; ; p=1; q=-11.576; r=16.35 ; ; D = q**2 - 4pr = 11.576**2 - 4 * 1 * 16.35 = 68.6129741827 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.58 ± sqrt{ 68.61 } }{ 2 } ; ; c_{1,2} = 5.78819864 ± 4.14164744343 ; ; c_{1} = 9.92984608343 ; ;
c_{2} = 1.64655119657 ; ; ; ; text{ Factored form: } ; ; (c -9.92984608343) (c -1.64655119657) = 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 = 6.2 ; ; b = 4.7 ; ; c = 9.93 ; ;

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

p = a+b+c = 6.2+4.7+9.93 = 20.83 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.83 }{ 2 } = 10.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.41 * (10.41-6.2)(10.41-4.7)(10.41-9.93) } ; ; T = sqrt{ 121.69 } = 11.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.03 }{ 6.2 } = 3.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.03 }{ 4.7 } = 4.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.03 }{ 9.93 } = 2.22 ; ;

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{ 4.7**2+9.93**2-6.2**2 }{ 2 * 4.7 * 9.93 } ) = 28° 12'44" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.2**2+9.93**2-4.7**2 }{ 2 * 6.2 * 9.93 } ) = 21° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 6.2**2+4.7**2-9.93**2 }{ 2 * 6.2 * 4.7 } ) = 130° 47'16" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.03 }{ 10.41 } = 1.06 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.2 }{ 2 * sin 28° 12'44" } = 6.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.7**2+2 * 9.93**2 - 6.2**2 } }{ 2 } = 7.123 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.93**2+2 * 6.2**2 - 4.7**2 } }{ 2 } = 7.937 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.7**2+2 * 6.2**2 - 9.93**2 } }{ 2 } = 2.37 ; ;







#2 Obtuse scalene triangle.

Sides: a = 6.2   b = 4.7   c = 1.64765512009

Area: T = 1.82992206508
Perimeter: p = 12.54765512009
Semiperimeter: s = 6.27332756004

Angle ∠ A = α = 151.7887651159° = 151°47'16″ = 2.64991942766 rad
Angle ∠ B = β = 21° = 0.36765191429 rad
Angle ∠ C = γ = 7.21223488407° = 7°12'44″ = 0.12658792341 rad

Height: ha = 0.59900711777
Height: hb = 0.77883917663
Height: hc = 2.22218812872

Median: ma = 1.67704985569
Median: mb = 3.88798280153
Median: mc = 5.43994133218

Inradius: r = 0.29215893972
Circumradius: R = 6.55875060576

Vertex coordinates: A[1.64765512009; 0] B[0; 0] C[5.78881986443; 2.22218812872]
Centroid: CG[2.47882499484; 0.74106270957]
Coordinates of the circumscribed circle: U[0.82332756004; 6.50656208759]
Coordinates of the inscribed circle: I[1.57332756004; 0.29215893972]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 28.21223488407° = 28°12'44″ = 2.64991942766 rad
∠ B' = β' = 159° = 0.36765191429 rad
∠ C' = γ' = 172.7887651159° = 172°47'16″ = 0.12658792341 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 6.2 ; ; b = 4.7 ; ; beta = 21° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 4.7**2 = 6.2**2 + c**2 -2 * 6.2 * c * cos (21° ) ; ; ; ; c**2 -11.576c +16.35 =0 ; ; p=1; q=-11.576; r=16.35 ; ; D = q**2 - 4pr = 11.576**2 - 4 * 1 * 16.35 = 68.6129741827 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.58 ± sqrt{ 68.61 } }{ 2 } ; ; c_{1,2} = 5.78819864 ± 4.14164744343 ; ; c_{1} = 9.92984608343 ; ; : Nr. 1
c_{2} = 1.64655119657 ; ; ; ; text{ Factored form: } ; ; (c -9.92984608343) (c -1.64655119657) = 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 = 6.2 ; ; b = 4.7 ; ; c = 1.65 ; ;

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

p = a+b+c = 6.2+4.7+1.65 = 12.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.55 }{ 2 } = 6.27 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.27 * (6.27-6.2)(6.27-4.7)(6.27-1.65) } ; ; T = sqrt{ 3.35 } = 1.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.83 }{ 6.2 } = 0.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.83 }{ 4.7 } = 0.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.83 }{ 1.65 } = 2.22 ; ;

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{ 4.7**2+1.65**2-6.2**2 }{ 2 * 4.7 * 1.65 } ) = 151° 47'16" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.2**2+1.65**2-4.7**2 }{ 2 * 6.2 * 1.65 } ) = 21° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 6.2**2+4.7**2-1.65**2 }{ 2 * 6.2 * 4.7 } ) = 7° 12'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.83 }{ 6.27 } = 0.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.2 }{ 2 * sin 151° 47'16" } = 6.56 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.7**2+2 * 1.65**2 - 6.2**2 } }{ 2 } = 1.67 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.65**2+2 * 6.2**2 - 4.7**2 } }{ 2 } = 3.88 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.7**2+2 * 6.2**2 - 1.65**2 } }{ 2 } = 5.439 ; ;
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