Triangle calculator SSA

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Triangle has two solutions with side c=67.02770155446 and with side c=3.894395228

#1 Obtuse scalene triangle.

Sides: a = 45   b = 42   c = 67.02770155446

Area: T = 928.484390373
Perimeter: p = 154.0277015545
Semiperimeter: s = 77.01435077723

Angle ∠ A = α = 41.27222167114° = 41°16'20″ = 0.72203360712 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 100.7287783289° = 100°43'40″ = 1.75880314666 rad

Height: ha = 41.26659512769
Height: hb = 44.21435192252
Height: hc = 27.70547663897

Median: ma = 51.20660582979
Median: mb = 53.08330519696
Median: mc = 27.77330948365

Inradius: r = 12.05661175641
Circumradius: R = 34.11096541551

Vertex coordinates: A[67.02770155446; 0] B[0; 0] C[35.46604839123; 27.70547663897]
Centroid: CG[34.1622499819; 9.23549221299]
Coordinates of the circumscribed circle: U[33.51435077723; -6.34992758152]
Coordinates of the inscribed circle: I[35.01435077723; 12.05661175641]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.7287783289° = 138°43'40″ = 0.72203360712 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 79.27222167114° = 79°16'20″ = 1.75880314666 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 42 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 45**2 + c**2 -2 * 45 * c * cos (38° ) ; ; ; ; c**2 -70.921c +261 =0 ; ; p=1; q=-70.921; r=261 ; ; D = q**2 - 4pr = 70.921**2 - 4 * 1 * 261 = 3985.78367718 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 70.92 ± sqrt{ 3985.78 } }{ 2 } ; ;
c_{1,2} = 35.46048391 ± 31.5665316323 ; ; c_{1} = 67.0270155446 ; ; c_{2} = 3.89395227998 ; ; ; ; text{ Factored form: } ; ; (c -67.0270155446) (c -3.89395227998) = 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 = 45 ; ; b = 42 ; ; c = 67.03 ; ;

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

p = a+b+c = 45+42+67.03 = 154.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 154.03 }{ 2 } = 77.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77.01 * (77.01-45)(77.01-42)(77.01-67.03) } ; ; T = sqrt{ 862082.36 } = 928.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 928.48 }{ 45 } = 41.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 928.48 }{ 42 } = 44.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 928.48 }{ 67.03 } = 27.7 ; ;

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{ 42**2+67.03**2-45**2 }{ 2 * 42 * 67.03 } ) = 41° 16'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+67.03**2-42**2 }{ 2 * 45 * 67.03 } ) = 38° ; ;
 gamma = 180° - alpha - beta = 180° - 41° 16'20" - 38° = 100° 43'40" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 928.48 }{ 77.01 } = 12.06 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 41° 16'20" } = 34.11 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 67.03**2 - 45**2 } }{ 2 } = 51.206 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 67.03**2+2 * 45**2 - 42**2 } }{ 2 } = 53.083 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 45**2 - 67.03**2 } }{ 2 } = 27.773 ; ;



#2 Obtuse scalene triangle.

Sides: a = 45   b = 42   c = 3.894395228

Area: T = 53.94105191246
Perimeter: p = 90.894395228
Semiperimeter: s = 45.447697614

Angle ∠ A = α = 138.7287783289° = 138°43'40″ = 2.42112565824 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 3.27222167114° = 3°16'20″ = 0.05771109555 rad

Height: ha = 2.39773564055
Height: hb = 2.56985961488
Height: hc = 27.70547663897

Median: ma = 19.57988516563
Median: mb = 24.06441108745
Median: mc = 43.48222870133

Inradius: r = 1.18768890673
Circumradius: R = 34.11096541551

Vertex coordinates: A[3.894395228; 0] B[0; 0] C[35.46604839123; 27.70547663897]
Centroid: CG[13.11881453974; 9.23549221299]
Coordinates of the circumscribed circle: U[1.947697614; 34.05440422049]
Coordinates of the inscribed circle: I[3.447697614; 1.18768890673]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.27222167114° = 41°16'20″ = 2.42112565824 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 176.7287783289° = 176°43'40″ = 0.05771109555 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 42 ; ; beta = 38° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 45**2 + c**2 -2 * 45 * c * cos (38° ) ; ; ; ; c**2 -70.921c +261 =0 ; ; p=1; q=-70.921; r=261 ; ; D = q**2 - 4pr = 70.921**2 - 4 * 1 * 261 = 3985.78367718 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 70.92 ± sqrt{ 3985.78 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 35.46048391 ± 31.5665316323 ; ; c_{1} = 67.0270155446 ; ; c_{2} = 3.89395227998 ; ; ; ; text{ Factored form: } ; ; (c -67.0270155446) (c -3.89395227998) = 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 = 45 ; ; b = 42 ; ; c = 3.89 ; ;

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

p = a+b+c = 45+42+3.89 = 90.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.89 }{ 2 } = 45.45 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.45 * (45.45-45)(45.45-42)(45.45-3.89) } ; ; T = sqrt{ 2909.58 } = 53.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.94 }{ 45 } = 2.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.94 }{ 42 } = 2.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.94 }{ 3.89 } = 27.7 ; ;

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{ 42**2+3.89**2-45**2 }{ 2 * 42 * 3.89 } ) = 138° 43'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+3.89**2-42**2 }{ 2 * 45 * 3.89 } ) = 38° ; ;
 gamma = 180° - alpha - beta = 180° - 138° 43'40" - 38° = 3° 16'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.94 }{ 45.45 } = 1.19 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 138° 43'40" } = 34.11 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 3.89**2 - 45**2 } }{ 2 } = 19.579 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.89**2+2 * 45**2 - 42**2 } }{ 2 } = 24.064 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 45**2 - 3.89**2 } }{ 2 } = 43.482 ; ;
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