Triangle calculator SSA

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Triangle has two solutions with side c=10.2665585662 and with side c=3.33003475024

#1 Acute scalene triangle.

Sides: a = 12.8   b = 11.4   c = 10.2665585662

Area: T = 55.71765464101
Perimeter: p = 34.4665585662
Semiperimeter: s = 17.2332792831

Angle ∠ A = α = 72.21221794948° = 72°12'44″ = 1.26603402922 rad
Angle ∠ B = β = 58° = 1.01222909662 rad
Angle ∠ C = γ = 49.78878205052° = 49°47'16″ = 0.86989613952 rad

Height: ha = 8.70657103766
Height: hb = 9.77548327035
Height: hc = 10.85550156308

Median: ma = 8.75884887105
Median: mb = 10.10554997151
Median: mc = 10.98797284918

Inradius: r = 3.23331698615
Circumradius: R = 6.72113168992

Vertex coordinates: A[10.2665585662; 0] B[0; 0] C[6.78329665822; 10.85550156308]
Centroid: CG[5.68328507481; 3.61883385436]
Coordinates of the circumscribed circle: U[5.1332792831; 4.33994168517]
Coordinates of the inscribed circle: I[5.8332792831; 3.23331698615]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.7887820505° = 107°47'16″ = 1.26603402922 rad
∠ B' = β' = 122° = 1.01222909662 rad
∠ C' = γ' = 130.2122179495° = 130°12'44″ = 0.86989613952 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 = 12.8 ; ; b = 11.4 ; ; c = 10.27 ; ;

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

p = a+b+c = 12.8+11.4+10.27 = 34.47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.47 }{ 2 } = 17.23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.23 * (17.23-12.8)(17.23-11.4)(17.23-10.27) } ; ; T = sqrt{ 3104.33 } = 55.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 55.72 }{ 12.8 } = 8.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 55.72 }{ 11.4 } = 9.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 55.72 }{ 10.27 } = 10.86 ; ;

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{ 12.8**2-11.4**2-10.27**2 }{ 2 * 11.4 * 10.27 } ) = 72° 12'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-12.8**2-10.27**2 }{ 2 * 12.8 * 10.27 } ) = 58° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.27**2-12.8**2-11.4**2 }{ 2 * 11.4 * 12.8 } ) = 49° 47'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 55.72 }{ 17.23 } = 3.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 72° 12'44" } = 6.72 ; ;





#2 Obtuse scalene triangle.

Sides: a = 12.8   b = 11.4   c = 3.33003475024

Area: T = 17.91326618627
Perimeter: p = 27.55003475024
Semiperimeter: s = 13.75501737512

Angle ∠ A = α = 107.7887820505° = 107°47'16″ = 1.88112523614 rad
Angle ∠ B = β = 58° = 1.01222909662 rad
Angle ∠ C = γ = 14.21221794948° = 14°12'44″ = 0.24880493261 rad

Height: ha = 2.7998853416
Height: hb = 3.14325722566
Height: hc = 10.85550156308

Median: ma = 5.4288272913
Median: mb = 7.40878436011
Median: mc = 12.00773696783

Inradius: r = 1.30327225828
Circumradius: R = 6.72113168992

Vertex coordinates: A[3.33003475024; 0] B[0; 0] C[6.78329665822; 10.85550156308]
Centroid: CG[3.36111046949; 3.61883385436]
Coordinates of the circumscribed circle: U[1.65501737512; 6.51655987791]
Coordinates of the inscribed circle: I[2.35501737512; 1.30327225828]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.21221794948° = 72°12'44″ = 1.88112523614 rad
∠ B' = β' = 122° = 1.01222909662 rad
∠ C' = γ' = 165.7887820505° = 165°47'16″ = 0.24880493261 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.8 ; ; b = 11.4 ; ; beta = 58° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 11.4**2 = 12.8**2 + c**2 -2 * 11.4 * c * cos (58° ) ; ; ; ; c**2 -13.566c +33.88 =0 ; ; p=1; q=-13.5659331644; r=33.88 ; ; D = q**2 - 4pr = 13.566**2 - 4 * 1 * 33.88 = 48.5145426202 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 13.57 ± sqrt{ 48.51 } }{ 2 } ; ; c_{1,2} = 6.78296658219 ± 3.48261907981 ; ;
c_{1} = 10.265585662 ; ; c_{2} = 3.30034750238 ; ; ; ; (c -10.265585662) (c -3.30034750238) = 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 = 12.8 ; ; b = 11.4 ; ; c = 3.3 ; ;

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

p = a+b+c = 12.8+11.4+3.3 = 27.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.5 }{ 2 } = 13.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.75 * (13.75-12.8)(13.75-11.4)(13.75-3.3) } ; ; T = sqrt{ 320.86 } = 17.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.91 }{ 12.8 } = 2.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.91 }{ 11.4 } = 3.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.91 }{ 3.3 } = 10.86 ; ;

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{ 12.8**2-11.4**2-3.3**2 }{ 2 * 11.4 * 3.3 } ) = 107° 47'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-12.8**2-3.3**2 }{ 2 * 12.8 * 3.3 } ) = 58° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.3**2-12.8**2-11.4**2 }{ 2 * 11.4 * 12.8 } ) = 14° 12'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.91 }{ 13.75 } = 1.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.8 }{ 2 * sin 107° 47'16" } = 6.72 ; ;




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