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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 32.8**2 = 39.5**2 + c**2 -2 * 32.8 * c * cos (35° 18') ; ; ; ; c**2 -64.475c +484.41 =0 ; ; p=1; q=-64.4748696163; 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} = 32.2374348082 ± 23.5550886861 ; ;
c_{1} = 55.7925234943 ; ; c_{2} = 8.68234612205 ; ; ; ; (c -55.7925234943) (c -8.68234612205) = 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 = 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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