Triangle calculator SSA

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Triangle has two solutions with side c=1.97698238147 and with side c=0.15500134163

#1 Obtuse scalene triangle.

Sides: a = 1.06   b = 0.91   c = 1.97698238147

Area: T = 0.01329368295
Perimeter: p = 3.94398238147
Semiperimeter: s = 1.97699119073

Angle ∠ A = α = 0.82770405207° = 0°49'37″ = 0.01444345801 rad
Angle ∠ B = β = 0.71° = 0°42'36″ = 0.01223918377 rad
Angle ∠ C = γ = 178.4632959479° = 178°27'47″ = 3.11547662358 rad

Height: ha = 0.02444091123
Height: hb = 0.02884325923
Height: hc = 0.01331350118

Median: ma = 1.44398794847
Median: mb = 1.51548854512
Median: mc = 0.0766148111

Inradius: r = 0.00765672122
Circumradius: R = 36.71986575882

Vertex coordinates: A[1.97698238147; 0] B[0; 0] C[1.06599186155; 0.01331350118]
Centroid: CG[1.01099141434; 0.00443783373]
Coordinates of the circumscribed circle: U[0.98549119073; -36.70554459667]
Coordinates of the inscribed circle: I[1.06599119073; 0.00765672122]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 179.1732959479° = 179°10'23″ = 0.01444345801 rad
∠ B' = β' = 179.29° = 179°17'24″ = 0.01223918377 rad
∠ C' = γ' = 1.53770405207° = 1°32'13″ = 3.11547662358 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 1.06 ; ; b = 0.91 ; ; beta = 0° 42'36" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 0.91**2 = 1.06**2 + c**2 -2 * 1.06 * c * cos (0° 42'36") ; ; ; ; c**2 -2.12c +0.296 =0 ; ; p=1; q=-2.12; r=0.296 ; ; D = q**2 - 4pr = 2.12**2 - 4 * 1 * 0.296 = 3.31170988586 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 2.12 ± sqrt{ 3.31 } }{ 2 } ; ; c_{1,2} = 1.05991862 ± 0.909905199164 ; ; c_{1} = 1.96982381916 ; ;
c_{2} = 0.150013420836 ; ; ; ; text{ Factored form: } ; ; (c -1.96982381916) (c -0.150013420836) = 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 = 1.06 ; ; b = 0.91 ; ; c = 1.97 ; ;

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

p = a+b+c = 1.06+0.91+1.97 = 3.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3.94 }{ 2 } = 1.97 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.97 * (1.97-1.06)(1.97-0.91)(1.97-1.97) } ; ; T = sqrt{ 0 } = 0.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.01 }{ 1.06 } = 0.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.01 }{ 0.91 } = 0.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.01 }{ 1.97 } = 0.01 ; ;

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{ 0.91**2+1.97**2-1.06**2 }{ 2 * 0.91 * 1.97 } ) = 0° 49'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.06**2+1.97**2-0.91**2 }{ 2 * 1.06 * 1.97 } ) = 0° 42'36" ; ; gamma = 180° - alpha - beta = 180° - 0° 49'37" - 0° 42'36" = 178° 27'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.01 }{ 1.97 } = 0.01 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.06 }{ 2 * sin 0° 49'37" } = 36.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 1.97**2 - 1.06**2 } }{ 2 } = 1.44 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.97**2+2 * 1.06**2 - 0.91**2 } }{ 2 } = 1.515 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 1.06**2 - 1.97**2 } }{ 2 } = 0.076 ; ;





#2 Obtuse scalene triangle.

Sides: a = 1.06   b = 0.91   c = 0.15500134163

Area: T = 0.0010985214
Perimeter: p = 2.12200134163
Semiperimeter: s = 1.06600067082

Angle ∠ A = α = 179.1732959479° = 179°10'23″ = 3.12771580735 rad
Angle ∠ B = β = 0.71° = 0°42'36″ = 0.01223918377 rad
Angle ∠ C = γ = 0.11770405207° = 0°7'1″ = 0.00220427424 rad

Height: ha = 0.00218588943
Height: hb = 0.00221653055
Height: hc = 0.01331350118

Median: ma = 0.38800026481
Median: mb = 0.60550016633
Median: mc = 0.98549994892

Inradius: r = 0.00109294413
Circumradius: R = 36.71986575881

Vertex coordinates: A[0.15500134163; 0] B[0; 0] C[1.06599186155; 0.01331350118]
Centroid: CG[0.40333106773; 0.00443783373]
Coordinates of the circumscribed circle: U[0.07550067082; 36.71985809785]
Coordinates of the inscribed circle: I[0.15500067082; 0.00109294413]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 0.82770405207° = 0°49'37″ = 3.12771580735 rad
∠ B' = β' = 179.29° = 179°17'24″ = 0.01223918377 rad
∠ C' = γ' = 179.8832959479° = 179°52'59″ = 0.00220427424 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 1.06 ; ; b = 0.91 ; ; beta = 0° 42'36" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 0.91**2 = 1.06**2 + c**2 -2 * 1.06 * c * cos (0° 42'36") ; ; ; ; c**2 -2.12c +0.296 =0 ; ; p=1; q=-2.12; r=0.296 ; ; D = q**2 - 4pr = 2.12**2 - 4 * 1 * 0.296 = 3.31170988586 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 2.12 ± sqrt{ 3.31 } }{ 2 } ; ; c_{1,2} = 1.05991862 ± 0.909905199164 ; ; c_{1} = 1.96982381916 ; ; : Nr. 1
c_{2} = 0.150013420836 ; ; ; ; text{ Factored form: } ; ; (c -1.96982381916) (c -0.150013420836) = 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 = 1.06 ; ; b = 0.91 ; ; c = 0.15 ; ;

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

p = a+b+c = 1.06+0.91+0.15 = 2.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.12 }{ 2 } = 1.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.06 * (1.06-1.06)(1.06-0.91)(1.06-0.15) } ; ; T = sqrt{ 0 } = 0 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0 }{ 1.06 } = 0 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0 }{ 0.91 } = 0 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0 }{ 0.15 } = 0.01 ; ;

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{ 0.91**2+0.15**2-1.06**2 }{ 2 * 0.91 * 0.15 } ) = 179° 10'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.06**2+0.15**2-0.91**2 }{ 2 * 1.06 * 0.15 } ) = 0° 42'36" ; ; gamma = 180° - alpha - beta = 180° - 179° 10'23" - 0° 42'36" = 0° 7'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0 }{ 1.06 } = 0 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.06 }{ 2 * sin 179° 10'23" } = 36.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 0.15**2 - 1.06**2 } }{ 2 } = 0.38 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.15**2+2 * 1.06**2 - 0.91**2 } }{ 2 } = 0.605 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 1.06**2 - 0.15**2 } }{ 2 } = 0.985 ; ;
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