Triangle calculator SSA

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Triangle has two solutions with side c=55.79325234943 and with side c=8.68223461221

#1 Obtuse scalene triangle.

Sides: a = 39.5   b = 32.8   c = 55.79325234943

Area: T = 636.7432667924
Perimeter: p = 128.0932523494
Semiperimeter: s = 64.04662617471

Angle ∠ A = α = 44.09986291593° = 44°5'55″ = 0.77696662744 rad
Angle ∠ B = β = 35.3° = 35°18' = 0.6166101226 rad
Angle ∠ C = γ = 100.6011370841° = 100°36'5″ = 1.75658251532 rad

Height: ha = 32.24401350848
Height: hb = 38.82657724344
Height: hc = 22.82553761631

Median: ma = 41.28326881263
Median: mb = 45.47105161498
Median: mc = 23.23545342225

Inradius: r = 9.94219177725
Circumradius: R = 28.3810693285

Vertex coordinates: A[55.79325234943; 0] B[0; 0] C[32.23774348082; 22.82553761631]
Centroid: CG[29.34333194341; 7.6088458721]
Coordinates of the circumscribed circle: U[27.89662617471; -5.22113342997]
Coordinates of the inscribed circle: I[31.24662617471; 9.94219177725]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.9011370841° = 135°54'5″ = 0.77696662744 rad
∠ B' = β' = 144.7° = 144°42' = 0.6166101226 rad
∠ C' = γ' = 79.39986291593° = 79°23'55″ = 1.75658251532 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 39.5 ; ; b = 32.8 ; ; beta = 35° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32.8**2 = 39.5**2 + c**2 -2 * 39.5 * c * cos (35° 18') ; ; ; ; c**2 -64.475c +484.41 =0 ; ; p=1; q=-64.475; r=484.41 ; ; D = q**2 - 4pr = 64.475**2 - 4 * 1 * 484.41 = 2219.36881204 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 64.47 ± sqrt{ 2219.37 } }{ 2 } ; ; c_{1,2} = 32.23743481 ± 23.5550886861 ; ;
c_{1} = 55.7925234961 ; ; c_{2} = 8.68234612388 ; ; ; ; (c -55.7925234961) (c -8.68234612388) = 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 = 39.5 ; ; b = 32.8 ; ; c = 55.79 ; ;

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

p = a+b+c = 39.5+32.8+55.79 = 128.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 128.09 }{ 2 } = 64.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.05 * (64.05-39.5)(64.05-32.8)(64.05-55.79) } ; ; T = sqrt{ 405441.23 } = 636.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 636.74 }{ 39.5 } = 32.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 636.74 }{ 32.8 } = 38.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 636.74 }{ 55.79 } = 22.83 ; ;

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{ 39.5**2-32.8**2-55.79**2 }{ 2 * 32.8 * 55.79 } ) = 44° 5'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.8**2-39.5**2-55.79**2 }{ 2 * 39.5 * 55.79 } ) = 35° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 55.79**2-39.5**2-32.8**2 }{ 2 * 32.8 * 39.5 } ) = 100° 36'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 636.74 }{ 64.05 } = 9.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.5 }{ 2 * sin 44° 5'55" } = 28.38 ; ;





#2 Obtuse scalene triangle.

Sides: a = 39.5   b = 32.8   c = 8.68223461221

Area: T = 99.08989081072
Perimeter: p = 80.98223461221
Semiperimeter: s = 40.4911173061

Angle ∠ A = α = 135.9011370841° = 135°54'5″ = 2.37219263791 rad
Angle ∠ B = β = 35.3° = 35°18' = 0.6166101226 rad
Angle ∠ C = γ = 8.79986291593° = 8°47'55″ = 0.15435650485 rad

Height: ha = 5.01771599042
Height: hb = 6.04220065919
Height: hc = 22.82553761631

Median: ma = 13.62216396624
Median: mb = 23.42876880441
Median: mc = 36.04444061742

Inradius: r = 2.44771730655
Circumradius: R = 28.3810693285

Vertex coordinates: A[8.68223461221; 0] B[0; 0] C[32.23774348082; 22.82553761631]
Centroid: CG[13.64399269767; 7.6088458721]
Coordinates of the circumscribed circle: U[4.3411173061; 28.04767104629]
Coordinates of the inscribed circle: I[7.6911173061; 2.44771730655]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 44.09986291593° = 44°5'55″ = 2.37219263791 rad
∠ B' = β' = 144.7° = 144°42' = 0.6166101226 rad
∠ C' = γ' = 171.2011370841° = 171°12'5″ = 0.15435650485 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 39.5 ; ; b = 32.8 ; ; beta = 35° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32.8**2 = 39.5**2 + c**2 -2 * 39.5 * c * cos (35° 18') ; ; ; ; c**2 -64.475c +484.41 =0 ; ; p=1; q=-64.475; r=484.41 ; ; D = q**2 - 4pr = 64.475**2 - 4 * 1 * 484.41 = 2219.36881204 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 64.47 ± sqrt{ 2219.37 } }{ 2 } ; ; c_{1,2} = 32.23743481 ± 23.5550886861 ; ; : Nr. 1
c_{1} = 55.7925234961 ; ; c_{2} = 8.68234612388 ; ; ; ; (c -55.7925234961) (c -8.68234612388) = 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 = 39.5 ; ; b = 32.8 ; ; c = 8.68 ; ;

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

p = a+b+c = 39.5+32.8+8.68 = 80.98 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80.98 }{ 2 } = 40.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.49 * (40.49-39.5)(40.49-32.8)(40.49-8.68) } ; ; T = sqrt{ 9818.61 } = 99.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 99.09 }{ 39.5 } = 5.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 99.09 }{ 32.8 } = 6.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 99.09 }{ 8.68 } = 22.83 ; ;

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{ 39.5**2-32.8**2-8.68**2 }{ 2 * 32.8 * 8.68 } ) = 135° 54'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.8**2-39.5**2-8.68**2 }{ 2 * 39.5 * 8.68 } ) = 35° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.68**2-39.5**2-32.8**2 }{ 2 * 32.8 * 39.5 } ) = 8° 47'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 99.09 }{ 40.49 } = 2.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.5 }{ 2 * sin 135° 54'5" } = 28.38 ; ;




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