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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 29.8**2 = 37.3**2 + c**2 -2 * 29.8 * c * cos (35° 24') ; ; ; ; c**2 -60.809c +503.25 =0 ; ; p=1; q=-60.8085335614; 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} = 30.4042667807 ± 20.5224130762 ; ;
c_{1} = 50.9266798569 ; ; c_{2} = 9.88185370447 ; ; ; ; (c -50.9266798569) (c -9.88185370447) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37.3**2-29.8**2-9.88**2 }{ 2 * 29.8 * 9.88 } ) = 133° 31'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29.8**2-37.3**2-9.88**2 }{ 2 * 37.3 * 9.88 } ) = 35° 24' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.88**2-37.3**2-29.8**2 }{ 2 * 29.8 * 37.3 } ) = 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 ; ;




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