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=38.3; b=33.1; c=6.68875556733 and a=38.3; b=33.1; c=55.51880424867.

#1 Obtuse scalene triangle.

Sides: a = 38.3   b = 33.1   c = 6.68875556733

Area: T = 74.73221920839
Perimeter: p = 78.08875556733
Semiperimeter: s = 39.04437778366

Angle ∠ A = α = 137.5299123672° = 137°31'45″ = 2.44003360255 rad
Angle ∠ B = β = 35.7° = 35°42' = 0.6233082543 rad
Angle ∠ C = γ = 6.77108763283° = 6°46'15″ = 0.11881740852 rad

Height: ha = 3.90224643386
Height: hb = 4.51655403072
Height: hc = 22.35496283949

Median: ma = 14.26333867101
Median: mb = 21.95223165165
Median: mc = 35.6388029544

Inradius: r = 1.91440615029
Circumradius: R = 28.36113216649

Vertex coordinates: A[6.68875556733; 0] B[0; 0] C[31.103279908; 22.35496283949]
Centroid: CG[12.59767849177; 7.45498761316]
Coordinates of the circumscribed circle: U[3.34437778366; 28.16435174714]
Coordinates of the inscribed circle: I[5.94437778366; 1.91440615029]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.47108763283° = 42°28'15″ = 2.44003360255 rad
∠ B' = β' = 144.3° = 144°18' = 0.6233082543 rad
∠ C' = γ' = 173.2299123672° = 173°13'45″ = 0.11881740852 rad




How did we calculate this triangle?

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

a = 38.3 ; ; b = 33.1 ; ; beta = 35.7° ; ;

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 ; ; ; ; 33.1**2 = 38.3**2 + c**2 - 2 * 38.3 * c * cos 35° 42' ; ; ; ; ; ; c**2 -62.206c +371.28 =0 ; ; p=1; q=-62.206; r=371.28 ; ; D = q**2 - 4pr = 62.206**2 - 4 * 1 * 371.28 = 2384.41644243 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.21 ± sqrt{ 2384.42 } }{ 2 } ; ; c_{1,2} = 31.10279908 ± 24.4152434067 ; ; c_{1} = 55.5180424867 ; ; c_{2} = 6.68755567331 ; ;
 ; ; text{ Factored form: } ; ; (c -55.5180424867) (c -6.68755567331) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.518 ; ;
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 = 38.3 ; ; b = 33.1 ; ; c = 6.69 ; ;

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

p = a+b+c = 38.3+33.1+6.69 = 78.09 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78.09 }{ 2 } = 39.04 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.04 * (39.04-38.3)(39.04-33.1)(39.04-6.69) } ; ; T = sqrt{ 5584.9 } = 74.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74.73 }{ 38.3 } = 3.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74.73 }{ 33.1 } = 4.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74.73 }{ 6.69 } = 22.35 ; ;

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{ 33.1**2+6.69**2-38.3**2 }{ 2 * 33.1 * 6.69 } ) = 137° 31'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.3**2+6.69**2-33.1**2 }{ 2 * 38.3 * 6.69 } ) = 35° 42' ; ; gamma = 180° - alpha - beta = 180° - 137° 31'45" - 35° 42' = 6° 46'15" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74.73 }{ 39.04 } = 1.91 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.3 }{ 2 * sin 137° 31'45" } = 28.36 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.1**2+2 * 6.69**2 - 38.3**2 } }{ 2 } = 14.263 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.69**2+2 * 38.3**2 - 33.1**2 } }{ 2 } = 21.952 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.1**2+2 * 38.3**2 - 6.69**2 } }{ 2 } = 35.638 ; ;







#2 Obtuse scalene triangle.

Sides: a = 38.3   b = 33.1   c = 55.51880424867

Area: T = 620.4043809396
Perimeter: p = 126.9188042487
Semiperimeter: s = 63.45990212433

Angle ∠ A = α = 42.47108763283° = 42°28'15″ = 0.74112566281 rad
Angle ∠ B = β = 35.7° = 35°42' = 0.6233082543 rad
Angle ∠ C = γ = 101.8299123672° = 101°49'45″ = 1.77772534825 rad

Height: ha = 32.39770657648
Height: hb = 37.48766350088
Height: hc = 22.35496283949

Median: ma = 41.54995062715
Median: mb = 44.72988388042
Median: mc = 22.59883791368

Inradius: r = 9.77664478122
Circumradius: R = 28.36113216649

Vertex coordinates: A[55.51880424867; 0] B[0; 0] C[31.103279908; 22.35496283949]
Centroid: CG[28.87436138555; 7.45498761316]
Coordinates of the circumscribed circle: U[27.75990212433; -5.81438890764]
Coordinates of the inscribed circle: I[30.35990212433; 9.77664478122]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.5299123672° = 137°31'45″ = 0.74112566281 rad
∠ B' = β' = 144.3° = 144°18' = 0.6233082543 rad
∠ C' = γ' = 78.17108763283° = 78°10'15″ = 1.77772534825 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 38.3 ; ; b = 33.1 ; ; beta = 35.7° ; ; : 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 ; ; ; ; 33.1**2 = 38.3**2 + c**2 - 2 * 38.3 * c * cos 35° 42' ; ; ; ; ; ; c**2 -62.206c +371.28 =0 ; ; p=1; q=-62.206; r=371.28 ; ; D = q**2 - 4pr = 62.206**2 - 4 * 1 * 371.28 = 2384.41644243 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.21 ± sqrt{ 2384.42 } }{ 2 } ; ; c_{1,2} = 31.10279908 ± 24.4152434067 ; ; c_{1} = 55.5180424867 ; ; c_{2} = 6.68755567331 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -55.5180424867) (c -6.68755567331) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.518 ; ; : 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 = 38.3 ; ; b = 33.1 ; ; c = 55.52 ; ;

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

p = a+b+c = 38.3+33.1+55.52 = 126.92 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 126.92 }{ 2 } = 63.46 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.46 * (63.46-38.3)(63.46-33.1)(63.46-55.52) } ; ; T = sqrt{ 384900.89 } = 620.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 620.4 }{ 38.3 } = 32.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 620.4 }{ 33.1 } = 37.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 620.4 }{ 55.52 } = 22.35 ; ;

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{ 33.1**2+55.52**2-38.3**2 }{ 2 * 33.1 * 55.52 } ) = 42° 28'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.3**2+55.52**2-33.1**2 }{ 2 * 38.3 * 55.52 } ) = 35° 42' ; ; gamma = 180° - alpha - beta = 180° - 42° 28'15" - 35° 42' = 101° 49'45" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 620.4 }{ 63.46 } = 9.78 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.3 }{ 2 * sin 42° 28'15" } = 28.36 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.1**2+2 * 55.52**2 - 38.3**2 } }{ 2 } = 41.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.52**2+2 * 38.3**2 - 33.1**2 } }{ 2 } = 44.729 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33.1**2+2 * 38.3**2 - 55.52**2 } }{ 2 } = 22.598 ; ;
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