Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle β.

Triangle has two solutions: a=37.9; b=26.4; c=18.21774715364 and a=37.9; b=26.4; c=40.59901553638.

#1 Obtuse scalene triangle.

Sides: a = 37.9   b = 26.4   c = 18.21774715364

Area: T = 217.8166091035
Perimeter: p = 82.51774715364
Semiperimeter: s = 41.25987357682

Angle ∠ A = α = 115.0769991308° = 115°4'12″ = 2.00883502186 rad
Angle ∠ B = β = 39.12° = 39°7'12″ = 0.68327728034 rad
Angle ∠ C = γ = 25.81100086922° = 25°48'36″ = 0.45504696316 rad

Height: ha = 11.49442528251
Height: hb = 16.50112190178
Height: hc = 23.91328784254

Median: ma = 12.46325693414
Median: mb = 26.64440074799
Median: mc = 31.36442460886

Inradius: r = 5.27992720615
Circumradius: R = 20.92109444007

Vertex coordinates: A[18.21774715364; 0] B[0; 0] C[29.40438134501; 23.91328784254]
Centroid: CG[15.87437616622; 7.97109594751]
Coordinates of the circumscribed circle: U[9.10987357682; 18.83439280906]
Coordinates of the inscribed circle: I[14.85987357682; 5.27992720615]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.93300086922° = 64°55'48″ = 2.00883502186 rad
∠ B' = β' = 140.88° = 140°52'48″ = 0.68327728034 rad
∠ C' = γ' = 154.1989991308° = 154°11'24″ = 0.45504696316 rad




How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 37.9 ; ; b = 26.4 ; ; beta = 39.12° ; ;

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 26.4**2 = 37.9**2 + c**2 - 2 * 37.9 * c * cos 39° 7'12" ; ; ; ; ; ; c**2 -58.808c +739.45 =0 ; ; p=1; q=-58.808; r=739.45 ; ; D = q**2 - 4pr = 58.808**2 - 4 * 1 * 739.45 = 500.53698164 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 58.81 ± sqrt{ 500.54 } }{ 2 } ; ; c_{1,2} = 29.40381345 ± 11.1863419137 ; ; c_{1} = 40.5901553637 ; ; c_{2} = 18.2174715363 ; ; ; ; text{ Factored form: } ; ; (c -40.5901553637) (c -18.2174715363) = 0 ; ; ; ; c > 0 ; ; ; ; c = 40.59 ; ;


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 = 37.9 ; ; b = 26.4 ; ; c = 18.22 ; ;

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

p = a+b+c = 37.9+26.4+18.22 = 82.52 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.52 }{ 2 } = 41.26 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.26 * (41.26-37.9)(41.26-26.4)(41.26-18.22) } ; ; T = sqrt{ 47443.85 } = 217.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 217.82 }{ 37.9 } = 11.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 217.82 }{ 26.4 } = 16.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 217.82 }{ 18.22 } = 23.91 ; ;

7. 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{ 26.4**2+18.22**2-37.9**2 }{ 2 * 26.4 * 18.22 } ) = 115° 4'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.9**2+18.22**2-26.4**2 }{ 2 * 37.9 * 18.22 } ) = 39° 7'12" ; ; gamma = 180° - alpha - beta = 180° - 115° 4'12" - 39° 7'12" = 25° 48'36" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 217.82 }{ 41.26 } = 5.28 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.9 }{ 2 * sin 115° 4'12" } = 20.92 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.4**2+2 * 18.22**2 - 37.9**2 } }{ 2 } = 12.463 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.22**2+2 * 37.9**2 - 26.4**2 } }{ 2 } = 26.644 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.4**2+2 * 37.9**2 - 18.22**2 } }{ 2 } = 31.364 ; ;







#2 Acute scalene triangle.

Sides: a = 37.9   b = 26.4   c = 40.59901553638

Area: T = 485.3143725243
Perimeter: p = 104.8990155364
Semiperimeter: s = 52.44550776819

Angle ∠ A = α = 64.93300086922° = 64°55'48″ = 1.1333242435 rad
Angle ∠ B = β = 39.12° = 39°7'12″ = 0.68327728034 rad
Angle ∠ C = γ = 75.95499913078° = 75°57' = 1.32655774152 rad

Height: ha = 25.61102229679
Height: hb = 36.76661913062
Height: hc = 23.91328784254

Median: ma = 28.51659228543
Median: mb = 36.98330414681
Median: mc = 25.5898958984

Inradius: r = 9.25437516711
Circumradius: R = 20.92109444007

Vertex coordinates: A[40.59901553638; 0] B[0; 0] C[29.40438134501; 23.91328784254]
Centroid: CG[23.3311322938; 7.97109594751]
Coordinates of the circumscribed circle: U[20.29550776819; 5.07989503348]
Coordinates of the inscribed circle: I[26.04550776819; 9.25437516711]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.0769991308° = 115°4'12″ = 1.1333242435 rad
∠ B' = β' = 140.88° = 140°52'48″ = 0.68327728034 rad
∠ C' = γ' = 104.0550008692° = 104°3' = 1.32655774152 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 37.9 ; ; b = 26.4 ; ; beta = 39.12° ; ; : Nr. 1

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 26.4**2 = 37.9**2 + c**2 - 2 * 37.9 * c * cos 39° 7'12" ; ; ; ; ; ; c**2 -58.808c +739.45 =0 ; ; p=1; q=-58.808; r=739.45 ; ; D = q**2 - 4pr = 58.808**2 - 4 * 1 * 739.45 = 500.53698164 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 58.81 ± sqrt{ 500.54 } }{ 2 } ; ; c_{1,2} = 29.40381345 ± 11.1863419137 ; ; c_{1} = 40.5901553637 ; ; c_{2} = 18.2174715363 ; ; ; ; text{ Factored form: } ; ; (c -40.5901553637) (c -18.2174715363) = 0 ; ; ; ; c > 0 ; ; ; ; c = 40.59 ; ; : 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 = 37.9 ; ; b = 26.4 ; ; c = 40.59 ; ;

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

p = a+b+c = 37.9+26.4+40.59 = 104.89 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 104.89 }{ 2 } = 52.45 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52.45 * (52.45-37.9)(52.45-26.4)(52.45-40.59) } ; ; T = sqrt{ 235529.41 } = 485.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 485.31 }{ 37.9 } = 25.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 485.31 }{ 26.4 } = 36.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 485.31 }{ 40.59 } = 23.91 ; ;

7. 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{ 26.4**2+40.59**2-37.9**2 }{ 2 * 26.4 * 40.59 } ) = 64° 55'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.9**2+40.59**2-26.4**2 }{ 2 * 37.9 * 40.59 } ) = 39° 7'12" ; ; gamma = 180° - alpha - beta = 180° - 64° 55'48" - 39° 7'12" = 75° 57' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 485.31 }{ 52.45 } = 9.25 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.9 }{ 2 * sin 64° 55'48" } = 20.92 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.4**2+2 * 40.59**2 - 37.9**2 } }{ 2 } = 28.516 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40.59**2+2 * 37.9**2 - 26.4**2 } }{ 2 } = 36.983 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.4**2+2 * 37.9**2 - 40.59**2 } }{ 2 } = 25.589 ; ;
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