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?

1. Use Law of Cosines

a = 105 ; ; b = 90 ; ; beta = 55° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 105**2 + c**2 -2 * 105 * c * cos (55° ) ; ; ; ; c**2 -120.451c +2925 =0 ; ; p=1; q=-120.451; 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} = 60.22552582 ± 26.4974330817 ; ; c_{1} = 86.7229589017 ; ; c_{2} = 33.7280927383 ; ; ; ; text{ Factored form: } ; ; (c -86.7229589017) (c -33.7280927383) = 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 = 86.72 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+86.72**2-105**2 }{ 2 * 90 * 86.72 } ) = 72° 52'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 105**2+86.72**2-90**2 }{ 2 * 105 * 86.72 } ) = 55° ; ; gamma = 180° - alpha - beta = 180° - 72° 52'39" - 55° = 52° 7'21" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 86.72**2 - 105**2 } }{ 2 } = 71.093 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 86.72**2+2 * 105**2 - 90**2 } }{ 2 } = 85.135 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 105**2 - 86.72**2 } }{ 2 } = 87.649 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 105 ; ; b = 90 ; ; beta = 55° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 105**2 + c**2 -2 * 105 * c * cos (55° ) ; ; ; ; c**2 -120.451c +2925 =0 ; ; p=1; q=-120.451; 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} = 60.22552582 ± 26.4974330817 ; ; c_{1} = 86.7229589017 ; ; c_{2} = 33.7280927383 ; ; ; ; text{ Factored form: } ; ; (c -86.7229589017) (c -33.7280927383) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 90**2+33.73**2-105**2 }{ 2 * 90 * 33.73 } ) = 107° 7'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 105**2+33.73**2-90**2 }{ 2 * 105 * 33.73 } ) = 55° ; ; gamma = 180° - alpha - beta = 180° - 107° 7'21" - 55° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 33.73**2 - 105**2 } }{ 2 } = 43.157 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.73**2+2 * 105**2 - 90**2 } }{ 2 } = 63.689 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 105**2 - 33.73**2 } }{ 2 } = 96.323 ; ;
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