Triangle calculator

You have entered side a, b and angle β.

Triangle has two solutions: a=38.9; b=32.1; c=8.74332847182 and a=38.9; b=32.1; c=55.2219521674.

#1 Obtuse scalene triangle.

Sides: a = 38.9 b = 32.1 c = 8.74332847182

Area: T = 96.81099040256
Perimeter: p = 79.74332847182
Semiperimeter: s = 39.87216423591

Angle ∠ A = α = 136.3879819764° = 136°22'47″ = 2.38802768882 rad
Angle ∠ B = β = 34.7° = 34°42' = 0.60656292504 rad
Angle ∠ C = γ = 8.92201802363° = 8°55'13″ = 0.1565686515 rad

Height: h_a = 4.97773729576
Height: h_b = 6.03217697212
Height: h_c = 22.14549734615

Median: m_a = 13.23334807905
Median: m_b = 23.17881149758
Median: m_c = 35.39334844722

Inradius: r = 2.42880390347
Circumradius: R = 28.19435311906

Vertex coordinates: A[8.74332847182; 0] B[0; 0] C[31.98114031961; 22.14549734615]
Centroid: CG[13.57548959714; 7.38216578205]
Coordinates of the circumscribed circle: U[4.37216423591; 27.85325392752]
Coordinates of the inscribed circle: I[7.77216423591; 2.42880390347]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 43.62201802363° = 43°37'13″ = 2.38802768882 rad
∠ B' = β' = 145.3° = 145°18' = 0.60656292504 rad
∠ C' = γ' = 171.0879819764° = 171°4'47″ = 0.1565686515 rad

How did we calculate this triangle?

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

$a = 38.9 ; ; b = 32.1 ; ; beta = 34.7° ; ;$

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

$b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 32.1**2 = 38.9**2 + c**2 - 2 * 38.9 * c * cos 34° 42' ; ; ; ; ; ; c**2 -63.963c +482.8 =0 ; ; p=1; q=-63.963; r=482.8 ; ; D = q**2 - 4pr = 63.963**2 - 4 * 1 * 482.8 = 2160.04060156 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.96 ± sqrt{ 2160.04 } }{ 2 } ; ;$
$c_{1,2} = 31.9814032 ± 23.2381184779 ; ; c_{1} = 55.219521674 ; ; c_{2} = 8.74328471823 ; ; ; ; text{ Factored form: } ; ; (c -55.219521674) (c -8.74328471823) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.22 ; ;$
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.

$a = 38.9 ; ; b = 32.1 ; ; c = 8.74 ; ;$

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

$p = a+b+c = 38.9+32.1+8.74 = 79.74 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 79.74 }{ 2 } = 39.87 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.87 * (39.87-38.9)(39.87-32.1)(39.87-8.74) } ; ; T = sqrt{ 9372.16 } = 96.81 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 96.81 }{ 38.9 } = 4.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 96.81 }{ 32.1 } = 6.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 96.81 }{ 8.74 } = 22.14 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 32.1**2+8.74**2-38.9**2 }{ 2 * 32.1 * 8.74 } ) = 136° 22'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.9**2+8.74**2-32.1**2 }{ 2 * 38.9 * 8.74 } ) = 34° 42' ; ;$
$gamma = 180° - alpha - beta = 180° - 136° 22'47" - 34° 42' = 8° 55'13" ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 96.81 }{ 39.87 } = 2.43 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.9 }{ 2 * sin 136° 22'47" } = 28.19 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.1**2+2 * 8.74**2 - 38.9**2 } }{ 2 } = 13.233 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.74**2+2 * 38.9**2 - 32.1**2 } }{ 2 } = 23.178 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.1**2+2 * 38.9**2 - 8.74**2 } }{ 2 } = 35.393 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 38.9 b = 32.1 c = 55.2219521674

Area: T = 611.4177421012
Perimeter: p = 126.2219521674
Semiperimeter: s = 63.1109760837

Angle ∠ A = α = 43.62201802363° = 43°37'13″ = 0.76113157654 rad
Angle ∠ B = β = 34.7° = 34°42' = 0.60656292504 rad
Angle ∠ C = γ = 101.6879819764° = 101°40'47″ = 1.77546476377 rad

Height: h_a = 31.43553429826
Height: h_b = 38.09545433652
Height: h_c = 22.14549734615

Median: m_a = 40.76215049642
Median: m_b = 44.98444449443
Median: m_c = 22.57223527025

Inradius: r = 9.68881593735
Circumradius: R = 28.19435311906

Vertex coordinates: A[55.2219521674; 0] B[0; 0] C[31.98114031961; 22.14549734615]
Centroid: CG[29.06769749567; 7.38216578205]
Coordinates of the circumscribed circle: U[27.6109760837; -5.70875658138]
Coordinates of the inscribed circle: I[31.0109760837; 9.68881593735]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.3879819764° = 136°22'47″ = 0.76113157654 rad
∠ B' = β' = 145.3° = 145°18' = 0.60656292504 rad
∠ C' = γ' = 78.32201802363° = 78°19'13″ = 1.77546476377 rad

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How did we calculate this triangle?

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

$a = 38.9 ; ; b = 32.1 ; ; beta = 34.7° ; ; : Nr. 1$

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

$b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 32.1**2 = 38.9**2 + c**2 - 2 * 38.9 * c * cos 34° 42' ; ; ; ; ; ; c**2 -63.963c +482.8 =0 ; ; p=1; q=-63.963; r=482.8 ; ; D = q**2 - 4pr = 63.963**2 - 4 * 1 * 482.8 = 2160.04060156 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.96 ± sqrt{ 2160.04 } }{ 2 } ; ; : Nr. 1$
$c_{1,2} = 31.9814032 ± 23.2381184779 ; ; c_{1} = 55.219521674 ; ; c_{2} = 8.74328471823 ; ; ; ; text{ Factored form: } ; ; (c -55.219521674) (c -8.74328471823) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.22 ; ; : Nr. 1$
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.

$a = 38.9 ; ; b = 32.1 ; ; c = 55.22 ; ;$

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

$p = a+b+c = 38.9+32.1+55.22 = 126.22 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 126.22 }{ 2 } = 63.11 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.11 * (63.11-38.9)(63.11-32.1)(63.11-55.22) } ; ; T = sqrt{ 373831.26 } = 611.42 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 611.42 }{ 38.9 } = 31.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 611.42 }{ 32.1 } = 38.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 611.42 }{ 55.22 } = 22.14 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 32.1**2+55.22**2-38.9**2 }{ 2 * 32.1 * 55.22 } ) = 43° 37'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.9**2+55.22**2-32.1**2 }{ 2 * 38.9 * 55.22 } ) = 34° 42' ; ;$
$gamma = 180° - alpha - beta = 180° - 43° 37'13" - 34° 42' = 101° 40'47" ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 611.42 }{ 63.11 } = 9.69 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.9 }{ 2 * sin 43° 37'13" } = 28.19 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.1**2+2 * 55.22**2 - 38.9**2 } }{ 2 } = 40.762 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.22**2+2 * 38.9**2 - 32.1**2 } }{ 2 } = 44.984 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.1**2+2 * 38.9**2 - 55.22**2 } }{ 2 } = 22.572 ; ;$
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