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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 45**2-42**2-67.03**2 }{ 2 * 42 * 67.03 } ) = 41° 16'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-45**2-67.03**2 }{ 2 * 45 * 67.03 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 67.03**2-45**2-42**2 }{ 2 * 42 * 45 } ) = 100° 43'40" ; ;

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 42**2 = 45**2 + c**2 -2 * 42 * c * cos (38° ) ; ; ; ; c**2 -70.921c +261 =0 ; ; p=1; q=-70.9209678246; 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} = 35.4604839123 ± 31.5665316323 ; ; c_{1} = 67.0270155446 ; ;
c_{2} = 3.89395227998 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 45**2-42**2-3.89**2 }{ 2 * 42 * 3.89 } ) = 138° 43'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-45**2-3.89**2 }{ 2 * 45 * 3.89 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.89**2-45**2-42**2 }{ 2 * 42 * 45 } ) = 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 ; ;




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