Triangle calculator SSA

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Triangle has two solutions with side c=50.92766798569 and with side c=9.88218537045

#1 Obtuse scalene triangle.

Sides: a = 37.3   b = 29.8   c = 50.92766798569

Area: T = 550.1911166025
Perimeter: p = 118.0276679857
Semiperimeter: s = 59.01333399284

Angle ∠ A = α = 46.47549553293° = 46°28'30″ = 0.81111409902 rad
Angle ∠ B = β = 35.4° = 35°24' = 0.61878465552 rad
Angle ∠ C = γ = 98.12550446707° = 98°7'30″ = 1.71326051082 rad

Height: ha = 29.5010866811
Height: hb = 36.92655816124
Height: hc = 21.60771877284

Median: ma = 37.3222390875
Median: mb = 42.07661020132
Median: mc = 22.16549119035

Inradius: r = 9.32331660281
Circumradius: R = 25.72215333613

Vertex coordinates: A[50.92766798569; 0] B[0; 0] C[30.40442667807; 21.60771877284]
Centroid: CG[27.11103155459; 7.20223959095]
Coordinates of the circumscribed circle: U[25.46333399284; -3.63553264154]
Coordinates of the inscribed circle: I[29.21333399284; 9.32331660281]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.5255044671° = 133°31'30″ = 0.81111409902 rad
∠ B' = β' = 144.6° = 144°36' = 0.61878465552 rad
∠ C' = γ' = 81.87549553293° = 81°52'30″ = 1.71326051082 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 37.3 ; ; b = 29.8 ; ; beta = 35° 24' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 29.8**2 = 37.3**2 + c**2 -2 * 37.3 * c * cos (35° 24') ; ; ; ; c**2 -60.809c +503.25 =0 ; ; p=1; q=-60.809; r=503.25 ; ; D = q**2 - 4pr = 60.809**2 - 4 * 1 * 503.25 = 1684.67775388 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.81 ± sqrt{ 1684.68 } }{ 2 } ; ; c_{1,2} = 30.40426678 ± 20.5224130762 ; ;
c_{1} = 50.9266798562 ; ; c_{2} = 9.88185370379 ; ; ; ; text{ Factored form: } ; ; (c -50.9266798562) (c -9.88185370379) = 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 = 37.3 ; ; b = 29.8 ; ; c = 50.93 ; ;

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

p = a+b+c = 37.3+29.8+50.93 = 118.03 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 118.03 }{ 2 } = 59.01 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 59.01 * (59.01-37.3)(59.01-29.8)(59.01-50.93) } ; ; T = sqrt{ 302710.32 } = 550.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 550.19 }{ 37.3 } = 29.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 550.19 }{ 29.8 } = 36.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 550.19 }{ 50.93 } = 21.61 ; ;

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{ 29.8**2+50.93**2-37.3**2 }{ 2 * 29.8 * 50.93 } ) = 46° 28'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.3**2+50.93**2-29.8**2 }{ 2 * 37.3 * 50.93 } ) = 35° 24' ; ; gamma = 180° - alpha - beta = 180° - 46° 28'30" - 35° 24' = 98° 7'30" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 550.19 }{ 59.01 } = 9.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.3 }{ 2 * sin 46° 28'30" } = 25.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.8**2+2 * 50.93**2 - 37.3**2 } }{ 2 } = 37.322 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 50.93**2+2 * 37.3**2 - 29.8**2 } }{ 2 } = 42.076 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.8**2+2 * 37.3**2 - 50.93**2 } }{ 2 } = 22.165 ; ;







#2 Obtuse scalene triangle.

Sides: a = 37.3   b = 29.8   c = 9.88218537045

Area: T = 106.7659534048
Perimeter: p = 76.98218537045
Semiperimeter: s = 38.49109268522

Angle ∠ A = α = 133.5255044671° = 133°31'30″ = 2.33304516634 rad
Angle ∠ B = β = 35.4° = 35°24' = 0.61878465552 rad
Angle ∠ C = γ = 11.07549553293° = 11°4'30″ = 0.1933294435 rad

Height: ha = 5.72443717988
Height: hb = 7.16550693992
Height: hc = 21.60771877284

Median: ma = 12.04325502415
Median: mb = 22.85773952216
Median: mc = 33.39553925241

Inradius: r = 2.77436285608
Circumradius: R = 25.72215333613

Vertex coordinates: A[9.88218537045; 0] B[0; 0] C[30.40442667807; 21.60771877284]
Centroid: CG[13.42987068284; 7.20223959095]
Coordinates of the circumscribed circle: U[4.94109268522; 25.24325141438]
Coordinates of the inscribed circle: I[8.69109268522; 2.77436285608]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 46.47549553293° = 46°28'30″ = 2.33304516634 rad
∠ B' = β' = 144.6° = 144°36' = 0.61878465552 rad
∠ C' = γ' = 168.9255044671° = 168°55'30″ = 0.1933294435 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 37.3 ; ; b = 29.8 ; ; beta = 35° 24' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 29.8**2 = 37.3**2 + c**2 -2 * 37.3 * c * cos (35° 24') ; ; ; ; c**2 -60.809c +503.25 =0 ; ; p=1; q=-60.809; r=503.25 ; ; D = q**2 - 4pr = 60.809**2 - 4 * 1 * 503.25 = 1684.67775388 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.81 ± sqrt{ 1684.68 } }{ 2 } ; ; c_{1,2} = 30.40426678 ± 20.5224130762 ; ; : Nr. 1
c_{1} = 50.9266798562 ; ; c_{2} = 9.88185370379 ; ; ; ; text{ Factored form: } ; ; (c -50.9266798562) (c -9.88185370379) = 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 = 37.3 ; ; b = 29.8 ; ; c = 9.88 ; ;

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

p = a+b+c = 37.3+29.8+9.88 = 76.98 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.98 }{ 2 } = 38.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.49 * (38.49-37.3)(38.49-29.8)(38.49-9.88) } ; ; T = sqrt{ 11397.6 } = 106.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.76 }{ 37.3 } = 5.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.76 }{ 29.8 } = 7.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.76 }{ 9.88 } = 21.61 ; ;

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{ 29.8**2+9.88**2-37.3**2 }{ 2 * 29.8 * 9.88 } ) = 133° 31'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.3**2+9.88**2-29.8**2 }{ 2 * 37.3 * 9.88 } ) = 35° 24' ; ; gamma = 180° - alpha - beta = 180° - 133° 31'30" - 35° 24' = 11° 4'30" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.76 }{ 38.49 } = 2.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.3 }{ 2 * sin 133° 31'30" } = 25.72 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.8**2+2 * 9.88**2 - 37.3**2 } }{ 2 } = 12.043 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.88**2+2 * 37.3**2 - 29.8**2 } }{ 2 } = 22.857 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.8**2+2 * 37.3**2 - 9.88**2 } }{ 2 } = 33.395 ; ;
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