Triangle calculator SSA

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Triangle has two solutions with side c=60.30884679844 and with side c=15.12222544774

#1 Obtuse scalene triangle.

Sides: a = 44   b = 32   c = 60.30884679844

Area: T = 683.3455459532
Perimeter: p = 136.3088467984
Semiperimeter: s = 68.15442339922

Angle ∠ A = α = 45.08768125955° = 45°5'13″ = 0.7876913329 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 103.9133187405° = 103°54'47″ = 1.81436272565 rad

Height: ha = 31.06111572515
Height: hb = 42.70990912208
Height: hc = 22.6621675296

Median: ma = 42.97215679876
Median: mb = 50.30546285675
Median: mc = 23.89897922206

Inradius: r = 10.02664564577
Circumradius: R = 31.06656644226

Vertex coordinates: A[60.30884679844; 0] B[0; 0] C[37.71553612309; 22.6621675296]
Centroid: CG[32.67546097384; 7.55438917653]
Coordinates of the circumscribed circle: U[30.15442339922; -7.47697843582]
Coordinates of the inscribed circle: I[36.15442339922; 10.02664564577]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.9133187405° = 134°54'47″ = 0.7876913329 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 76.08768125955° = 76°5'13″ = 1.81436272565 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 44 ; ; b = 32 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 44**2 + c**2 -2 * 44 * c * cos (31° ) ; ; ; ; c**2 -75.431c +912 =0 ; ; p=1; q=-75.431; r=912 ; ; D = q**2 - 4pr = 75.431**2 - 4 * 1 * 912 = 2041.79389111 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 75.43 ± sqrt{ 2041.79 } }{ 2 } ; ; c_{1,2} = 37.71536123 ± 22.5931067535 ; ; c_{1} = 60.3084679835 ; ; c_{2} = 15.1222544765 ; ; ; ; text{ Factored form: } ; ; (c -60.3084679835) (c -15.1222544765) = 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 = 44 ; ; b = 32 ; ; c = 60.31 ; ;

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

p = a+b+c = 44+32+60.31 = 136.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 136.31 }{ 2 } = 68.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 68.15 * (68.15-44)(68.15-32)(68.15-60.31) } ; ; T = sqrt{ 466961.02 } = 683.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 683.35 }{ 44 } = 31.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 683.35 }{ 32 } = 42.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 683.35 }{ 60.31 } = 22.66 ; ;

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{ 32**2+60.31**2-44**2 }{ 2 * 32 * 60.31 } ) = 45° 5'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 44**2+60.31**2-32**2 }{ 2 * 44 * 60.31 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 45° 5'13" - 31° = 103° 54'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 683.35 }{ 68.15 } = 10.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 44 }{ 2 * sin 45° 5'13" } = 31.07 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 60.31**2 - 44**2 } }{ 2 } = 42.972 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60.31**2+2 * 44**2 - 32**2 } }{ 2 } = 50.305 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 44**2 - 60.31**2 } }{ 2 } = 23.89 ; ;







#2 Obtuse scalene triangle.

Sides: a = 44   b = 32   c = 15.12222544774

Area: T = 171.3487810355
Perimeter: p = 91.12222544774
Semiperimeter: s = 45.56111272387

Angle ∠ A = α = 134.9133187405° = 134°54'47″ = 2.35546793246 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 14.08768125955° = 14°5'13″ = 0.24658612609 rad

Height: ha = 7.78985368343
Height: hb = 10.70992381472
Height: hc = 22.6621675296

Median: ma = 11.9310686914
Median: mb = 28.74661526163
Median: mc = 37.72204103223

Inradius: r = 3.76108334284
Circumradius: R = 31.06656644226

Vertex coordinates: A[15.12222544774; 0] B[0; 0] C[37.71553612309; 22.6621675296]
Centroid: CG[17.61325385694; 7.55438917653]
Coordinates of the circumscribed circle: U[7.56111272387; 30.13114596543]
Coordinates of the inscribed circle: I[13.56111272387; 3.76108334284]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 45.08768125955° = 45°5'13″ = 2.35546793246 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 165.9133187405° = 165°54'47″ = 0.24658612609 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 44 ; ; b = 32 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 44**2 + c**2 -2 * 44 * c * cos (31° ) ; ; ; ; c**2 -75.431c +912 =0 ; ; p=1; q=-75.431; r=912 ; ; D = q**2 - 4pr = 75.431**2 - 4 * 1 * 912 = 2041.79389111 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 75.43 ± sqrt{ 2041.79 } }{ 2 } ; ; c_{1,2} = 37.71536123 ± 22.5931067535 ; ; c_{1} = 60.3084679835 ; ; c_{2} = 15.1222544765 ; ; ; ; text{ Factored form: } ; ; (c -60.3084679835) (c -15.1222544765) = 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 = 44 ; ; b = 32 ; ; c = 15.12 ; ;

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

p = a+b+c = 44+32+15.12 = 91.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 91.12 }{ 2 } = 45.56 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.56 * (45.56-44)(45.56-32)(45.56-15.12) } ; ; T = sqrt{ 29360.07 } = 171.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 171.35 }{ 44 } = 7.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 171.35 }{ 32 } = 10.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 171.35 }{ 15.12 } = 22.66 ; ;

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{ 32**2+15.12**2-44**2 }{ 2 * 32 * 15.12 } ) = 134° 54'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 44**2+15.12**2-32**2 }{ 2 * 44 * 15.12 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 134° 54'47" - 31° = 14° 5'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 171.35 }{ 45.56 } = 3.76 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 44 }{ 2 * sin 134° 54'47" } = 31.07 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 15.12**2 - 44**2 } }{ 2 } = 11.931 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.12**2+2 * 44**2 - 32**2 } }{ 2 } = 28.746 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 44**2 - 15.12**2 } }{ 2 } = 37.72 ; ;
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