Triangle calculator

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

You have entered side a, b and angle γ.

Obtuse scalene triangle.

Sides: a = 45.3   b = 36.1   c = 74.37109776173

Area: T = 610.5499909318
Perimeter: p = 155.7710977617
Semiperimeter: s = 77.88554888087

Angle ∠ A = α = 27.0511019082° = 27°3'4″ = 0.4722129349 rad
Angle ∠ B = β = 21.2498980918° = 21°14'56″ = 0.37108646797 rad
Angle ∠ C = γ = 131.7° = 131°42' = 2.29985986249 rad

Height: ha = 26.95436383805
Height: hb = 33.82327096575
Height: hc = 16.41876921933

Median: ma = 53.89897360903
Median: mb = 58.87107368382
Median: mc = 17.17223446874

Inradius: r = 7.83884294515
Circumradius: R = 49.80438938952

Vertex coordinates: A[74.37109776173; 0] B[0; 0] C[42.22202484958; 16.41876921933]
Centroid: CG[38.86437420377; 5.47325640644]
Coordinates of the circumscribed circle: U[37.18554888087; -33.13110619991]
Coordinates of the inscribed circle: I[41.78554888087; 7.83884294515]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.9498980918° = 152°56'56″ = 0.4722129349 rad
∠ B' = β' = 158.7511019082° = 158°45'4″ = 0.37108646797 rad
∠ C' = γ' = 48.3° = 48°18' = 2.29985986249 rad

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How did we calculate this triangle?

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

a = 45.3 ; ; b = 36.1 ; ; gamma = 131.7° ; ;

2. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 45.3**2+36.1**2 - 2 * 45.3 * 36.1 * cos(131° 42') } ; ; c = 74.37 ; ;


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 = 45.3 ; ; b = 36.1 ; ; c = 74.37 ; ;

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

p = a+b+c = 45.3+36.1+74.37 = 155.77 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 155.77 }{ 2 } = 77.89 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77.89 * (77.89-45.3)(77.89-36.1)(77.89-74.37) } ; ; T = sqrt{ 372710.14 } = 610.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 610.5 }{ 45.3 } = 26.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 610.5 }{ 36.1 } = 33.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 610.5 }{ 74.37 } = 16.42 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 45.3**2-36.1**2-74.37**2 }{ 2 * 36.1 * 74.37 } ) = 27° 3'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 36.1**2-45.3**2-74.37**2 }{ 2 * 45.3 * 74.37 } ) = 21° 14'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 74.37**2-45.3**2-36.1**2 }{ 2 * 36.1 * 45.3 } ) = 131° 42' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 610.5 }{ 77.89 } = 7.84 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45.3 }{ 2 * sin 27° 3'4" } = 49.8 ; ;




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