Triangle calculator SSA

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Triangle has two solutions with side c=86.72329588985 and with side c=33.72880927352

#1 Acute scalene triangle.

Sides: a = 105   b = 90   c = 86.72329588985

Area: T = 3729.56326761
Perimeter: p = 281.7232958898
Semiperimeter: s = 140.8611479449

Angle ∠ A = α = 72.87774831208° = 72°52'39″ = 1.2721952031 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 52.12325168792° = 52°7'21″ = 0.9109709534 rad

Height: ha = 71.03992890685
Height: hb = 82.87991705799
Height: hc = 86.01109646503

Median: ma = 71.09327971039
Median: mb = 85.13548095673
Median: mc = 87.64986286257

Inradius: r = 26.47768103436
Circumradius: R = 54.93548564943

Vertex coordinates: A[86.72329588985; 0] B[0; 0] C[60.22655258169; 86.01109646503]
Centroid: CG[48.98328282385; 28.67703215501]
Coordinates of the circumscribed circle: U[43.36114794493; 33.72986311317]
Coordinates of the inscribed circle: I[50.86114794493; 26.47768103436]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.1232516879° = 107°7'21″ = 1.2721952031 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 127.8777483121° = 127°52'39″ = 0.9109709534 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 = 105 ; ; b = 90 ; ; c = 86.72 ; ;

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

p = a+b+c = 105+90+86.72 = 281.72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 281.72 }{ 2 } = 140.86 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 140.86 * (140.86-105)(140.86-90)(140.86-86.72) } ; ; T = sqrt{ 13909637.75 } = 3729.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3729.56 }{ 105 } = 71.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3729.56 }{ 90 } = 82.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3729.56 }{ 86.72 } = 86.01 ; ;

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{ 105**2-90**2-86.72**2 }{ 2 * 90 * 86.72 } ) = 72° 52'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-105**2-86.72**2 }{ 2 * 105 * 86.72 } ) = 55° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 86.72**2-105**2-90**2 }{ 2 * 90 * 105 } ) = 52° 7'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3729.56 }{ 140.86 } = 26.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 72° 52'39" } = 54.93 ; ;





#2 Obtuse scalene triangle.

Sides: a = 105   b = 90   c = 33.72880927352

Area: T = 1450.493289599
Perimeter: p = 228.7288092735
Semiperimeter: s = 114.3644046368

Angle ∠ A = α = 107.1232516879° = 107°7'21″ = 1.87696406226 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 17.87774831208° = 17°52'39″ = 0.31220209424 rad

Height: ha = 27.6288436114
Height: hb = 32.23331754663
Height: hc = 86.01109646503

Median: ma = 43.1577179238
Median: mb = 63.68990266826
Median: mc = 96.32329149274

Inradius: r = 12.68331197571
Circumradius: R = 54.93548564943

Vertex coordinates: A[33.72880927352; 0] B[0; 0] C[60.22655258169; 86.01109646503]
Centroid: CG[31.31878728507; 28.67703215501]
Coordinates of the circumscribed circle: U[16.86440463676; 52.28223335187]
Coordinates of the inscribed circle: I[24.36440463676; 12.68331197571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.87774831208° = 72°52'39″ = 1.87696406226 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 162.1232516879° = 162°7'21″ = 0.31220209424 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 105 ; ; b = 90 ; ; beta = 55° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 105**2 + c**2 -2 * 90 * c * cos (55° ) ; ; ; ; c**2 -120.451c +2925 =0 ; ; p=1; q=-120.451051634; r=2925 ; ; D = q**2 - 4pr = 120.451**2 - 4 * 1 * 2925 = 2808.45583967 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 120.45 ± sqrt{ 2808.46 } }{ 2 } ; ; c_{1,2} = 60.2255258169 ± 26.4974330817 ; ; c_{1} = 86.7229588985 ; ;
c_{2} = 33.7280927352 ; ; ; ; (c -86.7229588985) (c -33.7280927352) = 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 = 105 ; ; b = 90 ; ; c = 33.73 ; ;

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

p = a+b+c = 105+90+33.73 = 228.73 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 228.73 }{ 2 } = 114.36 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 114.36 * (114.36-105)(114.36-90)(114.36-33.73) } ; ; T = sqrt{ 2103929.64 } = 1450.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1450.49 }{ 105 } = 27.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1450.49 }{ 90 } = 32.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1450.49 }{ 33.73 } = 86.01 ; ;

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{ 105**2-90**2-33.73**2 }{ 2 * 90 * 33.73 } ) = 107° 7'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-105**2-33.73**2 }{ 2 * 105 * 33.73 } ) = 55° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 33.73**2-105**2-90**2 }{ 2 * 90 * 105 } ) = 17° 52'39" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1450.49 }{ 114.36 } = 12.68 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 107° 7'21" } = 54.93 ; ;




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