Triangle calculator SSA

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Triangle has two solutions with side c=9.06878427113 and with side c=2.01334336888

#1 Acute scalene triangle.

Sides: a = 8.92   b = 7.83   c = 9.06878427113

Area: T = 31.6954584162
Perimeter: p = 25.81878427113
Semiperimeter: s = 12.90989213557

Angle ∠ A = α = 63.22659572605° = 63°13'33″ = 1.10435011269 rad
Angle ∠ B = β = 51.6° = 51°36' = 0.9010589894 rad
Angle ∠ C = γ = 65.17440427395° = 65°10'27″ = 1.13875016326 rad

Height: ha = 7.10664090049
Height: hb = 8.0965679224
Height: hc = 6.99105456393

Median: ma = 7.20224812196
Median: mb = 8.097746014
Median: mc = 7.0632662893

Inradius: r = 2.45552465143
Circumradius: R = 4.99655757106

Vertex coordinates: A[9.06878427113; 0] B[0; 0] C[5.54106382001; 6.99105456393]
Centroid: CG[4.86994936371; 2.33301818798]
Coordinates of the circumscribed circle: U[4.53439213557; 2.09774588962]
Coordinates of the inscribed circle: I[5.07989213557; 2.45552465143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.774404274° = 116°46'27″ = 1.10435011269 rad
∠ B' = β' = 128.4° = 128°24' = 0.9010589894 rad
∠ C' = γ' = 114.826595726° = 114°49'33″ = 1.13875016326 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 = 8.92 ; ; b = 7.83 ; ; c = 9.07 ; ;

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

p = a+b+c = 8.92+7.83+9.07 = 25.82 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.82 }{ 2 } = 12.91 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.91 * (12.91-8.92)(12.91-7.83)(12.91-9.07) } ; ; T = sqrt{ 1004.55 } = 31.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.69 }{ 8.92 } = 7.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.69 }{ 7.83 } = 8.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.69 }{ 9.07 } = 6.99 ; ;

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{ 8.92**2-7.83**2-9.07**2 }{ 2 * 7.83 * 9.07 } ) = 63° 13'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.83**2-8.92**2-9.07**2 }{ 2 * 8.92 * 9.07 } ) = 51° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.07**2-8.92**2-7.83**2 }{ 2 * 7.83 * 8.92 } ) = 65° 10'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.69 }{ 12.91 } = 2.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.92 }{ 2 * sin 63° 13'33" } = 5 ; ;





#2 Obtuse scalene triangle.

Sides: a = 8.92   b = 7.83   c = 2.01334336888

Area: T = 7.03875000468
Perimeter: p = 18.76334336888
Semiperimeter: s = 9.38217168444

Angle ∠ A = α = 116.774404274° = 116°46'27″ = 2.03880915267 rad
Angle ∠ B = β = 51.6° = 51°36' = 0.9010589894 rad
Angle ∠ C = γ = 11.62659572605° = 11°37'33″ = 0.20329112329 rad

Height: ha = 1.57879148087
Height: hb = 1.79875734475
Height: hc = 6.99105456393

Median: ma = 3.57662840505
Median: mb = 5.1466157072
Median: mc = 8.33221168496

Inradius: r = 0.75501292315
Circumradius: R = 4.99655757106

Vertex coordinates: A[2.01334336888; 0] B[0; 0] C[5.54106382001; 6.99105456393]
Centroid: CG[2.5188023963; 2.33301818798]
Coordinates of the circumscribed circle: U[1.00767168444; 4.89330867431]
Coordinates of the inscribed circle: I[1.55217168444; 0.75501292315]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.22659572605° = 63°13'33″ = 2.03880915267 rad
∠ B' = β' = 128.4° = 128°24' = 0.9010589894 rad
∠ C' = γ' = 168.374404274° = 168°22'27″ = 0.20329112329 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.92 ; ; b = 7.83 ; ; beta = 51° 36' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 7.83**2 = 8.92**2 + c**2 -2 * 7.83 * c * cos (51° 36') ; ; ; ; c**2 -11.081c +18.258 =0 ; ; p=1; q=-11.0812764002; r=18.2575 ; ; D = q**2 - 4pr = 11.081**2 - 4 * 1 * 18.258 = 49.7646866569 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.08 ± sqrt{ 49.76 } }{ 2 } ; ; c_{1,2} = 5.54063820008 ± 3.52720451125 ; ;
c_{1} = 9.06784271134 ; ; c_{2} = 2.01343368883 ; ; ; ; (c -9.06784271134) (c -2.01343368883) = 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 = 8.92 ; ; b = 7.83 ; ; c = 2.01 ; ;

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

p = a+b+c = 8.92+7.83+2.01 = 18.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.76 }{ 2 } = 9.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.38 * (9.38-8.92)(9.38-7.83)(9.38-2.01) } ; ; T = sqrt{ 49.53 } = 7.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.04 }{ 8.92 } = 1.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.04 }{ 7.83 } = 1.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.04 }{ 2.01 } = 6.99 ; ;

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{ 8.92**2-7.83**2-2.01**2 }{ 2 * 7.83 * 2.01 } ) = 116° 46'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.83**2-8.92**2-2.01**2 }{ 2 * 8.92 * 2.01 } ) = 51° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.01**2-8.92**2-7.83**2 }{ 2 * 7.83 * 8.92 } ) = 11° 37'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.04 }{ 9.38 } = 0.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.92 }{ 2 * sin 116° 46'27" } = 5 ; ;




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