Triangle calculator SSA

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Triangle has two solutions with side c=46.96598413094 and with side c=1.34215718248

#1 Obtuse scalene triangle.

Sides: a = 32   b = 31   c = 46.96598413094

Area: T = 492.9354846256
Perimeter: p = 109.9659841309
Semiperimeter: s = 54.98799206547

Angle ∠ A = α = 42.62769572784° = 42°37'37″ = 0.74439807546 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 96.37330427216° = 96°22'23″ = 1.68220269057 rad

Height: ha = 30.8088427891
Height: hb = 31.80222481455
Height: hc = 20.99438889277

Median: ma = 36.43295669464
Median: mb = 37.07224068264
Median: mc = 21.00546024968

Inradius: r = 8.96657249481
Circumradius: R = 23.62659228439

Vertex coordinates: A[46.96598413094; 0] B[0; 0] C[24.15107065671; 20.99438889277]
Centroid: CG[23.70435159588; 6.99879629759]
Coordinates of the circumscribed circle: U[23.48799206547; -2.62325095379]
Coordinates of the inscribed circle: I[23.98799206547; 8.96657249481]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.3733042722° = 137°22'23″ = 0.74439807546 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 83.62769572784° = 83°37'37″ = 1.68220269057 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 = 32 ; ; b = 31 ; ; c = 46.96 ; ;

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

p = a+b+c = 32+31+46.96 = 109.96 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 109.96 }{ 2 } = 54.98 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.98 * (54.98-32)(54.98-31)(54.98-46.96) } ; ; T = sqrt{ 242984.76 } = 492.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 492.93 }{ 32 } = 30.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 492.93 }{ 31 } = 31.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 492.93 }{ 46.96 } = 20.99 ; ;

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{ 32**2-31**2-46.96**2 }{ 2 * 31 * 46.96 } ) = 42° 37'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-32**2-46.96**2 }{ 2 * 32 * 46.96 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.96**2-32**2-31**2 }{ 2 * 31 * 32 } ) = 96° 22'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 492.93 }{ 54.98 } = 8.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 42° 37'37" } = 23.63 ; ;





#2 Obtuse scalene triangle.

Sides: a = 32   b = 31   c = 1.34215718248

Area: T = 14.08224049397
Perimeter: p = 64.34215718248
Semiperimeter: s = 32.17107859124

Angle ∠ A = α = 137.3733042722° = 137°22'23″ = 2.3987611899 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 1.62769572784° = 1°37'37″ = 0.02883957613 rad

Height: ha = 0.88801503087
Height: hb = 0.90985422542
Height: hc = 20.99438889277

Median: ma = 15.01333243314
Median: mb = 16.51221139616
Median: mc = 31.49768259712

Inradius: r = 0.43877389156
Circumradius: R = 23.62659228439

Vertex coordinates: A[1.34215718248; 0] B[0; 0] C[24.15107065671; 20.99438889277]
Centroid: CG[8.49774261307; 6.99879629759]
Coordinates of the circumscribed circle: U[0.67107859124; 23.61663984656]
Coordinates of the inscribed circle: I[1.17107859124; 0.43877389156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.62769572784° = 42°37'37″ = 2.3987611899 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 178.3733042722° = 178°22'23″ = 0.02883957613 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 32 ; ; b = 31 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 31**2 = 32**2 + c**2 -2 * 31 * c * cos (41° ) ; ; ; ; c**2 -48.301c +63 =0 ; ; p=1; q=-48.3014131343; r=63 ; ; D = q**2 - 4pr = 48.301**2 - 4 * 1 * 63 = 2081.02651077 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 48.3 ± sqrt{ 2081.03 } }{ 2 } ; ; c_{1,2} = 24.1507065671 ± 22.8091347423 ; ; c_{1} = 46.9598413094 ; ;
c_{2} = 1.34157182485 ; ; ; ; (c -46.9598413094) (c -1.34157182485) = 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 = 32 ; ; b = 31 ; ; c = 1.34 ; ;

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

p = a+b+c = 32+31+1.34 = 64.34 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64.34 }{ 2 } = 32.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.17 * (32.17-32)(32.17-31)(32.17-1.34) } ; ; T = sqrt{ 198.31 } = 14.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.08 }{ 32 } = 0.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.08 }{ 31 } = 0.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.08 }{ 1.34 } = 20.99 ; ;

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{ 32**2-31**2-1.34**2 }{ 2 * 31 * 1.34 } ) = 137° 22'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-32**2-1.34**2 }{ 2 * 32 * 1.34 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.34**2-32**2-31**2 }{ 2 * 31 * 32 } ) = 1° 37'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.08 }{ 32.17 } = 0.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 137° 22'23" } = 23.63 ; ;




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